Polynomial interpolation of operators

Journal of Mathematical Sciences, VoL 86, No. 1, 1997

P O L Y N O M I A L I N T E R P O L A T I O N OF O P E R A T O R S * V. L . M a k a r o v a n d V. V. K h l o b y s t o v UDC 517.977.55

Necessary and sufficient conditions for the solvability of the polynomial operator interpolation problem in an arbitrary vector space are obtained (for the existence of a Hermite-type operator polynomial, conditions are obtained in a Hilbert space), lnterpolational operator formulas describing the whole set of intevpolants in these spaces as well as a subset of those polynomials preserving operator polynomials of the corresponding degree are constructed. In the metric of a measure space of operators~ an accuracy estimate is obtained and a theorem on the convergence of interpolational operator processes is proved for polynomial operators. Applications of the operator interpolation to the solution of some problems are described. Bibliography: 134 titles.

CHAPTER 2. A CLASS OF INTERPOLATIONAL OPERATOR FORMULAS,

THE HERM1TE-TYPE INTERPOLATION, AND INTERPOLATION IN ARBITRARY VECTOR SPACES

w I n t r o d u c t i o n The results presented in this chapter are an extension of the results obtained in the previous chapter.

In w a class of interpoIational operator formulas related to generalized matrices F - inverse to the matrix F is studied in a Hilbert space. Interpolational operator formula (5.2) from Chapter 1 is shown to be a representative of this class with F - = F + (F + is the Moore-Penrose matrix pseudoinverse to the matrix 1"). The necessary and sufficient condition for the solvability of the polynomial operator interpolation is obtained in terms of properties of some matrix whose elements depend on coordinates of linearly independent eigenvectors of the matr ix F corresponding to the zero eigenvalue. These conditions and conditions (4.1) from Chapter 1 are shown to be equivalent. The whole set of operator interpolants in a Hilbert space for the interpolational formula related to the generalized inverse matrix as well as its subset of c-interpolants is considered.

In w an extension of the operator interpolation to the case of the Hermite interpolation is given. Note that [129], in which a nonconstructive proof of the existence of the Hermite operator polynomial that transforms a Banach space into itself is obtained, is also devoted to these questions. Here the author restricts himself to the first Frech6t derivatives only and to the classical case n = 2m - 1 as well, where m is the number of nodes and n is the degree of the interpolational polynomial. The Hermite interpolational formula obtained in this paper is constructive but still does not possess the property of preserving an operator polynomial of the corresponding degree. In w of this chapter, the following results are obtained within the framework of Hilbert spaces for any m and n as well as for any orders of the Ggteaux differentials of the operator F at nodes along given directions. A class of the Hermite interpolational operator formulas related to generalized inverse matrices T - is constructed by using the form of interpolational formula (2.4). In so doing, in the case of T - = T + (T + is the pseudoinverse Moore-Penrose matrix) the Hermite interpolational formula in [46] is a representative of this class. Each of these formulas is shown to describe the whole set of Hermite operator polynomials in a Hilbert space, and a subset of the Hermite c-interpolants is described as well. The necessary and sufficient condition for the solvability of the Hermite operator interpolation problem is obtained in terms of properties of some matrix, the elements of which depend on coordinates of linearly independent eigenvectors of the matrix T corresponding to the zero eigenvalue. In conclusion, from the set of Hermite polynomials with T - = T + an interpolant that is the best approximation to F on this set in the metric of the Hilbert space H(0) is extracted.

*This paper is a continuation of the work published in Obchyslyuval'na ta Prykladna Matematyka, No. 78 (1994). The numeration of chapters, assertions, and formulas is continued.

Translated from Obchyslyuvallna ta Prykladna Matematyka, No. 79, 1995, pp. 10-116. Original article submitted November 5, 1994.

1072-3374/97/8601-2459518.00 �9 Plenum Publishing Corporation 2459

In w the principal results of construction of fundamentals of the general theory of polynomial operator interpolation (Chapter 1) are transferred to arbitrary vector spaces; thereby the requirements under which these results are valid are weakened. The main idea owing to which we manage to do this and to generalize

�9 lrn, results is that we replace the matrix F1 as the matrix F constructed from the system {x,}i=l in some Hilbert space by the matrix whose elements are linear functionals related to the linearly independent nodes x l , x2 , . . . ,zk, 1 < k <_ m. In this section the following is obtained. All the results of w obtained for the Hilbert spaces are extended to arbitrary vector spaces. Among these results are: the necessary and sufficient condition for the solvability of the polynomial operator interpolation problem, a class of interpolational operator formula.s, and a description of the whole set of interpolants as well as of its subset of c-interpolants in vector spaces. Here interpolational operator formulas are related, similarly to w to generalized inverse matrices F - , while the condition for the solvability of the interpolation problem is related to the matrix whose elements depend on coordinates of the linearly independent eigenvectors of the matrix F corresponding to the zero eigenvalue. Moreover, under certain conditions the original system of nodes is shown to generate a broader set of interpolational nodes, which is the union of this system and some span. This set, as well as the corresponding class of interpolational operator polynomials, the structure of which is invariant with respect to this set of nodes, is described. A theorem on c-interpolational operator polynomials at these nodes is presented. Note that a description of the whole set of operator interpolants of a given degree is obtained in this chapter; the results of [126-129] can be interpreted, to a certain extent, as a particular case of the results presented here.

w A class of i n t e r p o l a t i o n a l o p e r a t o r fo rmulas in a H i l b e r t space r e l a t e d to gene ra l i zed inverse m a t r i c e s

Consider the interpolational operator formula (5.2) from Chapter 1. The pseudoinverse Moore--Penrose matrix F + constructed for the matrix F is known to exist and to be unique. However, if we replace it in (5.2) by the generalized inverse matrix F - and rewrite the necessary and sufficient conditions (4.1) from Chapter 1 in terms of the matrix A0 = E - P -F , then formula (5.2) will be shown to remain the interpolational formula. But since the generalized inverse matrix F - is nonunique, we thereby obtain a class of interpolational operator formulas of the form (5.2) (Chapter 1). We show that in the necessary and sufficient condition for the solvability of the interpolational operator problem, the matrix acting at the vector -ff = (F(x l ) ,F (x2 ) , . . . ,F(xm)) depends only on eigenvectors of the matrix F corresponding to the zero eigenvalue.

Let us introduce the following notation. Let r = m - rkF; let 5i = (Cil,Ci2,... ,cim), i = 1 , . . . ,r, be linearly independent eigenvectors of the matrix

n m

r = xj) p = , i , j = l

corresponding to the zero eigenvalue, Fc'i = 0, i = 1 , . . . , r; and let

T h e o r e m 2.1. For the polynomial operator interpolation problem

P . ( x i ) = F ( x i ) , i = 1 , . ,m, Pn E KI.,

to be solvable, it is necessary and sufficient that the condition

zT=6, E - 0(llZ 'll) § IIZ tl)

(2.1)

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where E is the identity matrix of dimension (r x r), hold.

Proof. Necessity. Let the polynomial operator interpolation problem be solvable, i.e., let there exist the polynomial Q ' ( x ) E 17,, of n th degree such that ~t = ~}~, From the proof of Lemma 3.2 of Chapter 1, it follows that

= ci n = Q'(O)cie , i = 1 , . . . ,r,

o r

Z ~ = Z ~ " = Q' (O)Z~. (2.2)

If Z~'= 6, then Z ~ = O and condition (2.1) holds. Let Zg' 7~ I?. Then from (2.2) we find that

<z-P, z~') = Q'(o)IIZ~'II 2, (2.3)

Q'(o) = ( z ~ , z~)/llZ~ll ~.

Substituting (2.3) into (2.2), we get

z~ (z~,z~l _ z ~ z~(Z~)z~ = (~ z~(zr =-6 ~ p ~-~-~H ~ iiz~li 2 ) ,

which proves relation (2.1).

Sufficiency. Now let condition (2.1) hold. We show that in this ease the polynomial operator interpolation problem is always solvable. Note that if condition (2.1) holds, then it is also valid if the matrix Z in it is replaced by Z0 and the matrix E of dimension (r x r) is replaced by the matrix E of dimension ( m x m). It is not difficult to see that the inverse is true too. Consider the polynomial

(• " m) pin(X ) ~-- Q'(x)-~- - ~ ' , r - Z {(xi ,x)P}i=l ,

p = l

where Q,, (x) is an operator polynomial from the set

(2.4)

F - is a generalized inverse of the matrix F. Let us show that Pin(x) interpolates F at nodes xi, i = 1 , . . . , m. We have

Pin(xk) = Q'(zk) + ( ~ - U. , r-rr = Q'(zk) + ( ~ - -~]n,(E- Ao)gk) = F ( x ~ ) - ( A ; ~ - A;?]' , ~k).

Here A0 = E - F - F is an idempotent matrix. Indeed, taking into account the main relation for the generalized inverse matrix

r r - r = F,

we have A0 2 = ( E - F -F) 2 = E - 2F-F + F - F F - F = E - F - F = Ao.

Since an idempotent matrix is simple, it is similar to the diagonal matrix consisting of eigenvalues

(2.6)

Ao = X A X -1, (2.7)

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n ~ = P ' ( x ) + 5o(t lZr + JlZ~'Jl 2 - P ' ( O ) - : (2 .5)

where columns of the matr ix X can be interpreted as m linearly independent r ight-hand eigenvectors for A0 with the numbers 1 and 0 as corresponding eigenvalues. Here

A = d i a g { 1 , 1 , . . . , 1 , . , 0 ' 0 " " ' 0 } '

I" t l l - - 1"

X ~-- IIC',C'2""" C'r~r+,""" C'ml[;

c'i, i = 1 , . . . , m , are linearly independent right-hand eigenvalues of the matrix Ao. Note that a right- hand eigenvector of the matr ix r corresponding to the zero eigenvalue is the r ight-hand eigenveetor of the matrix A0 corresponding to the eigenvalue 1. Taking into account both (2.7) and matr ix equality AX' = Z0 from (2.6), we find

p l ( x k) : F(xk ) -- ( X t - 1 A X ' ~ - X ' - 1 A X t ~ n , g'k ) (2.8)

= F(x~)- (X'-' (Z0:~- Z0Z],), ~ ) = F(x~)- (X'-'(Zo~-Q,(O)Zog),~). = -~ r Z0Y = -~ is fulfilled, then p I ( x ) i s an If Z0g = 0 ~ Zg = 0' and the equality Z ] ~"

interpolational polynomial. Let Z o g r 0 ~ Z g~k ~. Then

z0Y-O.(O)z0g= ZoY <zY'z~ }jzgjl~> Zor -6 e r m, (2.9)

if equality (2.1) is valid. Thus, on the basis of (2.9) and (2.8), Phi (x) is again an interpolation operator polynomial, and hence the polynomial operator interpolation problem is solvable. The theorem is proved. []

Remark. It is not difficult to show that conditions (4.1) from Chapter 1 and (2.1) are equivalent. Indeed, let condition (4.1) from Chapter 1 hold, where A0 is a symmetric idempotent matrix. Then, by virtue of results of [32], a system of r ight-hand eigenvectors of the matrix A0 can be chosen to be the orthonormal system and

Ao = X A X ' , X ' X = X X ' = E.

Let Aog r {?. Then the equality

" ( X A X ' - - ~ , X A X ' g ) . . . . . ,_. ao~ :t _ (Ao-~,AOe) Aog= XAX,~: t iiAo~.ll 2 - ~ A I X A e (2.10)

=x(j (Zog, X'XZog) jZog= IIz0g[l' is valid.

Since the matrix X is nonsingular, we have that if the left-hand side of (2.10) is ~ E ym, then the parenthesized expression on the right-hand side of (2.10) is also the vector @ E ym. But as was noted above, Z0 and Z are interchangeable matrices in the sense that

Zof f=- '~ E y,,~ ~ Z f f=--~ E y,"

with ff E ym. Therefore, if for Aog • 0 condition (4.1) from Chapter 1 is fulfilled, then condition (2.1) holds as well.

Now let Aog = 0. Then

Ao~S:~= --~ E y,~ vV Z o ~ = - - ~ E y 'n e:~ Z ~=~=-~ E y , ".

It easy to show the inverse: if condition (2.1) holds, then condition (4.1) from Chapter 1 also takes place. Now let us proceed to the proof of a theorem for the class of interpolational operator formulas analogous

to the theorem from Section 1 of Chapter 1.

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T h e o r e m 2.2. Let conditior~ (2.1) hold and let F- be some generalized matrix inverse to the matrix F. Then in the case where Q , (x ) runs through the set (2.5), formula (2.4) aescribes the whole set of interpo-

�9 1 7 1 lational operator polynomials of degree n for F at interpolational nodes {x,}i=l C X.

Proof. The interpolationality of the polynomial defined by formula (2.4) under condition (2.1) follows from the proof of sufficiency (2.1). Now let P~ be some given interpolational operator polynomial for F of the nth degree at interpolation nodes {Xi}iml C X. Let us show that it belongs to the set (2.4). Indeed, taking into account the fact that P~ E II~ we set Q,(x) = P~ in (2.4) and immediately obtain P (x) • pO(x) The theorem is proved. []

So, let the necessary and sufficient condition for the solvability of the polynomial operator interpolation problem (2.1) hold. Then formula (2.4) describes the whole set of interpolants in the Hilbert space X. But since the matrix F - is nonunique, the expression (2.4) determines a class of interpolational operator formulas. It is easy to see that formula (5.2) from Chapter 1 with the Moore-Penrose matrix F + chosen as generalized matrix F - is a particular case of formula (2.4).

Note that by analogy with w of Chapter 1, it is possible to extract, from the set of interpolants (2.4), a subset of interpolants preserving a polynomial of the corresponding degree.

The following theorem is valid.

T h e o r e m 2.3. Let condition (2.1) hold and let in the definition of the set (2.5) I I , = II~(F) be the set of c-polynomials of degree n. Then formula (2.4) describes the whole set of c-interpolants of the nth degree in the Hilbert space X .

The proof completely repeats the proof of the corresponding Theorem 7 from Chapter 1.

w T h e H e r m l t e - t y p e i n t e r p o l a t i o n in a H i l b e r t space Consider the solution of the Hermite interpolation problem in the case where the values of the operator

and the first Ggteaux differentials along given directions are given at nodes. Thus, it is necessary to find a polynomial Pn E IIn satisfying the conditions

P n ( x i ) = F ( x i ) , PIn(Xi)Vi =F' (x i )v i , i = 1, . . . ,m. (3.1)

To solve the Hermite interpolation problem (3.1), let us proceed in the following manner. Consider the polynomial

/ _ ~p ~ 2 m P n ( x , a ) = Q n ( x ) + ~ - - ~ n , P-(o~)~-~.{~xi,x),~,=, , (3.2)

p = l

Q,(x) belongs to the set IIn and F - ( a ) is a generalized matrix inverse to the matrix n 2rn

r ( . ) = ; " p = l i , j= l

A0(c~) = E - r - ( ~ ) r ( , ) is an idempotent matrix;

~'1 ~ X l , X2 = X l -~- O~Vl~ ~'3 ~--- X2~

X2m = Xm Jr- O~Vm,

~'4 = X2 -~" O~V2, ' ' ' , ~ ' 2 m - - 1 = Xm~

a E R 1 , a # 0 .

Let us show that the limit of the polynomial P,~(x,a) as a --+ 0 exists and let us find a set H~ such that if Qn(x) E I I~ then this limiting polynomial is the Hermite interpolational polynomial satisfying conditions (3.1). Moreover, let us define a necessary and sufficient condition for the solvability of problem (3.1). Let us introduce the matrix

C(o ) =

1 0 . . . . . . . . . . . . . . . . . . 0 - -1/a 1/a 0 . . . . . . . . . . . . . . . 0

0 0 1 0 . . . . . . . . . . . . 0 0 0 - 1 / a 1/a 0 . . . . . . . . . 0

0 0 . . . . . . . . . . . . 0 t 0

0 0 . . . . . . . . . . . . 0 - -1 /a 1 / a

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T h e n 1 0 . . . . . . . . . . . . . . . . . . 0

1 a 0 . . . . . . . . . . . . . . . 0 0 0 1 0 . . . . . . . . . . . . 0

C -I(c~) = 0 0 1 ol 0 . . . . . . . . . 0

0 0 . . . . . . . . . . . . 0 1 0

0 0 . . . . . . . . . . . . 0 1

Let us t r a n s f o r m t h e s e c o n d a d d e n d in (3 .2) in the f o l l o w i n g m a n n e r :

_ p 2 m

p=l

p----1

Pu t c'-l(~)r-(~)c-l(~) = T - ( a ) . It is not difficult to see tha t the ma t r ix T-(c~) is a general ized inverse mat r ix to the m a t r i x T(c~) = c ( ~ ) r ( ~ ) C ' ( ~ ) .

Indeed, we have

T(c~)T-(o~)T(c~) = c(~)r(~)c'(~)c'-l(~)r-(~lC-'(~)C(~lr(~)c'(~) = c ( ~ ) r ( ~ ) r - ( ~ ) r ( ~ ) c ' ( ~ ) = c ( ~ ) r ( ~ ) c ' ( ~ ) = T(~),

s i n c e r ( ~ ) r - ( ~ ) r ( ~ ) = r ( ~ ) .

Let .A(oe) = E - T-(oe)T(oe) . Clearly, .A(c~) is an i d e m p o t e n t mat r ix :

A2(a) = ( E - T-(c~)T(c~)) 2 = E 2T-(c~)T(ce) + T-(o~)T(c~)T-(c~)T(c~) = E - T - ( o ~ ) T ( a ) = A ( a ) .

We represent the ma t r ix .A(c~) in the form

A(c~) = E - T-(o~)T(c~) = E - c'-l(~)r-(~)c-l(~)c(~)r(~)c'(~) = z - c ' - l ( ~ ) r - ( o ) r ( ~ ) c ' ( ~ ) = E - C ' - 1 ( . ) ( E - A0(~))C'(~) = C ' - I (~)A0(~)C ' (~) .

It is obvious tha t

l im C(cx)[P = ] ~ = ( F ( x l ) , F ' ( x l ) v l , F ( x 2 ) , F ' ( x 2 ) v 2 , . . . F ( x , , ) , F ' ( x m ) v m ) ; oz - + O

l im C(~)-O~. = -~n~ = (Qn(Xl),qtn(Xl)Vl,Qn(x2),QIn(x2)v2,...,Qn(xm),Qtn(Xm)Vm); o~--+0

l i m C ( ~ ) { ( 2 , i , x ) x ~ i = l = ( x l + , i = 0 , 1 c~ ---4 0 ~ - i t = 0 1----1 "

Now let us show tha t mat r ices T-(c~), T(c~), and A(c~) have l imits as c~ --+ 0. For this it is sufficient to show tha t the l imit T(c~) as c~ --+ 0 exists, since f rom the relat ions

T(c~)T- (o~)T(~) = T(c~), A(c~) = E - T-(c~)T(ce) (3.3)

the existence of l imits T-(c~) and A(c~) as c~ --+ 0 follows. Thus , let us find the l imit

l im T(c~) = l im C(cx)F(c~)C'(c~). oe--+0 oe---+0

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Without loss of generality, for the sake of simplicity we restrict ourselves to the case m = 2. Put

~(~,~) = ~ ( , , , v ) ~ , p=l

u , v ~ X .

Then the matr ix F(a) with m = 2 acquires the form

g(Xl ,Xl ) g(Xl + OtVl, Xl)

r(~) : 1 g ( ~ , ~ , ) I g(z2 + ave, x~)

g(xa,x~ + ~ ) g(Xl + aVl,Xl + OVl)

g(x2,Xl + ~Vl) g(x~ + av~,z~ + av~)

Let us represent the matr ix C ( a ) r ( a ) elementwise:

a ( x l , ~ ) g(Xl + O'UI,X2)

g(X2,X2) g(x~ + o~v2, z~)

g(Xl,X2 +~V2) g(xl + av~,x~ + ave)

g(x2,x2 + ~V2) g(z~ + av~,x: + av~)

~,~ = g(~,~,), .~: = g(~,x~ + ~ ) ,

a13 : g(xl ,x2) , a14 : g(xl ,x2 + av2),

g(Xl +C~Vl,Xl)--ff(Xl,2~l) a21 =

oc g(Xl + O~Vl~Xl + O~Vl) -- ff(Xl~Xl + OlYl)

a22 : Ol

a23 --~ o/

g(z~ + . . , , z~ + . , ~ ) - g(z , , ~2 + . ~ ) a 2 4 -~-

Ol a31 = g ( x 2 , x l ) , a32 = g(x2 , Xl + C~Vl),

g(.T2 + C~U2,Xl) -- g(X2,Xl) a41 =

o/ a(~: + . v : , ~ + ~ , l ) - g ( ~ , ~ +- ,~)

a,12 =

a,i4 =

ot

a43 = o~

Then we represent the matr ix C(a)F(a)C'(a) elementwise:

b22 =

b24 ~-

bl l = g ( ~ l , ~ l ) , b12 = g( x l , x l + ~ u 1 ) - g ( x l , X l ) ,

513 = g (Xl ,~2 ) , hi4 = g ( z i , x 2 @ ~U2) -- g (Xl ,X2) '

g(xl + ~ Y t , X l + ~ V l ) - - g ( x l , X l + ~ V l ) g(Xl + ~ V l , X l ) + f f ( X l , X l ) ~2 ~2

g(xi + ~v,,x2 + ~v2) - g(x~,x2 + a ~ ) g(xl + a~i,x2) + g(Xl,X2) ~2 ~2

b31 : g ( x 2 , x l ) , b32 = g (z2 ,Xl + ~ y l ) - f f ( x 2 , x l )

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b42 -~-

b44

' b33 = g(x2, x2), b34 = g(x2, x2 + ~v2) - g(x2, x2) ,

b~, = g ( ~ + " ~ ' ~ ' ) - g ( ~ ' ~ ' ) ,

g(x~ + ~ , ~ , + ~v,) - g ( ~ , ~ , + ~ , ) g(x~ +~v~ ,~ , ) + 9 ( ~ , ~ , ~2 ~2

g(x~ + ~v~, x~ + ~ ) - g (x~ ,~ + ~v~) g ( ~ + ~ , ~ ) + g ( ~ , ~ ~2 ~2

Evaluat ing l imits of e lements b,j as a --+ 0 by the L 'H6pi ta l rule, we finally wri te the ma t r i x

T = l im T ( a ) = l im C ( a ) r ( a ) C ' ( o ~ ) o~---40 c~--q.O

in the fo rm T = tls 2 l , s : l ,

where lit'~ll is a ma t r i x block of d imens ion (2 x 2):

If%lib,j=0,,, t~; = g(x,,x~),

= + , t~; = ~ g(x, + ~ , , x ~ ) , or=0

t ~ - O a o f l g ( x t + a v l , x ~ + f l v ~ ) , / , s = 1,2. a=~=o

Thus , it is not difficult to see tha t in the case of a rb i t ra ry m the ma t r i x T exists and can be represen ted in the form

T is m = lit [It,~=l' (3.4)

t ls Ii "tl = II , i l t , , j=o, , , = 1 , . . . ,m.

The ma t r ix T ob ta ined is a par t icu la r case of the mat r ix r cons t ruc ted in [46], where the Hermi te interpolat ional condi t ions conta in the Gs differentials at the nodes xz of orders up to ks (ks is an arbi t rary number ) a long direct ions t t t ~ 1 , ~ 2 , - . . , ~ i , i = 1 , . . . , k l .

Now on the basis of re la t ion (3.3) we can see tha t the l imits of the mat r ices T - (a) and `4(a) exist as a --+ 0. If we pu t

T - = l im T - ( a ) , Ct--40

and

then f rom rela t ion (3.3) it follows tha t

A = l im A(a), o~--~0

T T - T = T, A = E - T - T ,

and ,4 is an i d e m p o t e n t mat r ix . Let us in t roduce the set of opera to r polynomials of degree n of the form

+

~'~ = ( 1 , 0 , 1 , 0 , . . . ,1 ,0) E R2m

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and the operator polynomial of degree n:

, - ; : v _ , _ ( x , + ~ , , , x ) ~ , , (3.6) p = l ot=O I=1

where Q,, E II~ and T - is a generalized matrix inverse to the matrix T defined in (3.4). Further, by analogy with the results of Chapter 1, we get the necessary and sufficient condition for the solvability of the Hermite operator interpolation problem (3.1), and under these conditions we get the whole set of the Hermite polynomial operator interpolants of the nth degree.

Let us prove the following lemma beforehand.

L e m m a 3.1. The equality

.A'~,~z = .A'gzQ,~ (0) (3.7)

is valid.

Proof. On the basis of Lemma 3.2 of Chapter 1, we have

A o ( a ) ~ n = , ~ ' Ao(a)eQn(O).

We rewrite this relation as follows:

C ( a ) A ' o ( a ) C - t ( a ) C ( a ) ~ n = C(a)A~o(og)c-l(a)C(oe)gQn(O).

Passing to the limit as a -+ 0 on both the left-hand and right-hand sides of the last equality, and taking into account equalities

lira C(a)a~o(a)c- l (a) = lira .At(or) = .At c~--+O a--+O

and

we get (3.7). The lemma is proved. []

lim C(a)g = g~, o~-+0

T h e o r e m 3.1. For the Hermite operator interpolational problem to be solvable, that the condition

( E - ,A'g~ (,A'g~)' ) A, ~ : ~ - ~ E y2,~ a0 ( l l .a%ll) + [l~t'g~ll = =

hold, where E is the identity matrix of dimension (2m x 2m).

Proof. Necessity. Let the Hermite interpolational operator problem be solvable, i.e., let there exist the polynomial Qn(x) such that

Then, by virtue of Lemma 3.1, we have

it is necessary and sufficient

(3.8)

= = Q,~(O)A ~. (3.9)

Consider the following cases. (1) Let .A'g9 = 6. Then from (3.9) it follows that ,A'~:~ = O~' and condition (3.8) holds. (2) Now let A'g9 # {~. Then from (3.9) we find

( A ' - ~ , A%> = Qn(O)IIA'g~H ~ ,

Q.(O) : ( . 4 ' ~ , ~ t % ) (3.1o)

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Substituting (3.10) in (3.9), we get

A"~ (A"~3,A'g3) _ , . ( E _ A'E3(.A'E3)')A,-~3 =-~ E yem, - 11,,4.'----~%11 ~ ~ ' ~ = 11.,4.'6jl ~

and relation (3.8) holds again. Sufficiency. Let equality (3.8) be valid. Then we show that operator polynomial (3.6) satisfies the

Hermite interpolational conditions (3.1). Taking into account Lemma 3.1, we have

< { 19i t }m> e.(xk)=O~(xk)+ T ~ - ~ . ~ , 7"- ~ g ( ~ , + ~ v , , ~ ) , / = 0 , X c~=0 1=1

: F(Xk)- <r ~2k-l> : F(Xk) <r ~2k-1>, k = 1 , . . . , m .

( 3 . i 1 )

(1) Let .A'~'~ = 13. But then from conditions (3.8) it follows that A ' ~ = --~@, while from (3.11) we obtain Pn(xk) = F(xk). (2) Now let A'E9 7~ 0. But since Qn(x) E II~ we have

Q . ( o ) = (A'7~'A'r

Substituting this expression into (3.11), we find

(.4,3~ , x% ) .,~ P,~(xk) = F ( x k ) - .A'~:~ - ~,.~-,-5 ./-teg, s > = F(xk),

11-4%11 ,'

if condition (3.8) holds. For the first Gs differentials at the nodes Xk along directions vk, we have

ptn(Xk)V k I <0 { oi+l I }rn > = Q~(~)v~ + ~ - ~ , T - ggg3g(x~+ ,~ ,~ +~v~) , i=0,1 of=/5'=O /=1

= Q:(xk)vk + ( ~ - iffn~, T-TE2k> = O:(x~)vk + ( ~ : t - C]n~, ( E - A)E2k>

k = 1 , . . . , m ,

alld after that we repeat our previous reasoning. Thus, if condition (3.8) holds, then polynomial (3.6) satisfies the Hermite interpolational relations and hence the Hermite interpolational problem is solvable. The sufficiency of condition (3.8) and, hence, the theorem is proved. []

Now let us formulate a theorem analogous to Theorem 2.2 for the Hermite interpolation.

T h e o r e m 3.2. Let condition (3.8) hold and let T- be some generalized matrix inverse to the matrix T. Then in the case where Q,(x) runs through the set (3.5), formula (3.6) describes the whole set of Hermite interpolational operator polynomials of degree n for F satisfying conditions (3.1).

Proof. The interpolationality of polynomial (3.6) under condition (3.8) follows directly from the proof of the sufficiency of (3.8). Now let P~ be some given Hermite interpolational operator polynomial for F satisfying conditions (3.1). Let us show that it belongs to set (3.6). The latter is obvious since

2468

P~ E n ~ For the proof it is sufficient to put Q,,(x) = P~ in (3.6), and then we immediately get Pn I = P~ Thus, any Hermite interpolational polynomial satisfying conditions (3.1) belongs to the set (3.6). The theorem is proved. 123

Let the rank of the matrix T be equal to 2m - r. Let us show that the matrix acting at the vector ~ in (3.8) depends on linearly independent eigenvectors of the matrix T corresponding to the zero eigenvalue only. Indeed, .A is an idempotent and, hence, a simple matrix. But then it is representable in the form

A = x A x - ' , (3.12)

where X is a matrix, whose columns are orthonormal right-hand eigenvectors of the matrix .A corresponding to the eigenvalues

1, 1 , . . . ,1, 0 , 0 , . . . , 0 ,

r 2m--r

A : d i a g { ~ , l , . . . , 1 , 0 , 0 , . . . , 0 } . ~ ,

r 2m--r

Taking into account (3.12), we rewrite condition (3.8) with A'~'3 r 6 in the form

<r < X t - I A X t - ~ 9 , X t - I A X t ~ 9 > f l , ' ~ 9 - ~ - A ' g g = X ' - I A X ' ~ 9 - (X,_XAX,~9,X,_IAX,~9) X'-IAX'g9

= xt_aZo~:~ - <Z~176 x t _ X Z o ~ ~ ( Zo~9, ( X t X ) - l Zor )

Z~176 ZoT9 = "-~ e y2,-,, = X '-1 E - 1[Z0~.9[[2 ]

(3.13)

where E is the identi ty matrix of dimension (2m x 2m),

~ , , --' ---, ..,I T Zo ----- 1 2 " " c r O ~ �9

2m--r

c,, i = 1 , . . . , r, is an orthonormal system of vectors corresponding to eigenvectors of the matrix T with the zero eigenvalue (or corresponding to eigenvectors of the matrix .A with eigenvalue 1). But condition (3.13) is equivalent to the condition

E ( Z ~ ) ' } = �9 y~, = ~1~2- -c ; (3.14) Z ~ \

Z ~ Y T,

Z /

where E is the identity matrix of dimension (r x r). If, after all, A ' G = 6, then we rewrite condition (3.8) in the form

which, for its part, is equivalent to the condition

= r . (3.15)

2469

From (3.14) and (3.15) we get the necessary and sufficient condition for the solvability of the Hermite interpolation problem (3.1) in the form

E - / Z ~ = E r", (3.16) Z ~ (Z~)' \

~o (llZ~'~tl) + IIZ~'~ll ~ ,/

where Z = 11~1~2.-.~rllT and ~i, i = 1, . . . ,r, is an orthonormal system of eigenvectors of the matrix T corresponding to the zero eigenvalue.

Now we show that equality (3.16) is the necessary and sufficient condition for the solvability of the Hermite problem (3.1) in the case where c'i, i = 1 , . . . , r, is a linearly independent system of eigenvectors of the matrix T corresponding to the zero eigenvalue, as well. Thus, let the Hermite interpolational problem (3.1) be solvable, i.e., let there exist the polynomial Qn(x) such that i f 9 = ~,~9. In view of Lemma 3.1, we have

If A'~'~ = xt-lAXte9 = X'-IZo~3 = 0", then

z0~'~ : 6 ~ z~'~ = 6.

Hence

and condition (3.16) holds. If .x~'~ r 6 r Zo~'~ r 6 r z~'~ r ~, then

zT~= Z~Q.(e),

On(O)=(ZTg,Zg~) llZ~'~ll: '

z~(z~)'~ vr

and condition (3.16) hoids again. Thereby we proved the necessity of condition (3.16), where 6"i, i = 1 , . . . , r, is a linearly independent system of eigenvectors of the matrix T corresponding to the zero eigenvalue, for the solvability of the Hermite interpolational problem (3.1).

Now we prove the sufficiency of condition (3.16) in the case where c'i is a linearly independent system of vectors T~i = 0', i = 1 , . . . , r. To do this, we consider the set

H~ = P.(x) + ~o(llZ~'~ll) + IIZ~'~tl 2

and the operator polynomial (3.6), Qn E II~ It is not difficult to see that this polynomial satisfies conditions (3.1) if condition (3.16) holds (c~, i = 1 , . . . , r, are linearly independent). We have

Pn ( X k ) : Q n ( x k ) -}- ( ~:~r ~) - "~ n O , T - Te2 k - 1 ) : F (X k ) - ( ~z~l ~ 9 - ,Al ~ n ~ , C'2k--1)

~o (llZ~'~ll) + IIZ~'~ll 2 '

2470

But if condition (3.16) holds, the equality

z F ~ - ( z T ~ , z e ~ l z ~ = ~0 (llZGII) + IIZ~ll 2

is valid and P,,(xk) : F(xk), k = 1 , . . . ,m . With (3.16) being fulfilled, we prove by analogy that

P~(Xk)Vk : F'(xk)vk, k : 1 , . . . ,m .

Thereby the sufficiency of condition (3.16) with linearly independent system of vectors c'i, i = 1 , . . . , r, for the solvability of the Hermite interpolational problem (3.1) is shown. Thus, the following theorem is proved.

T h e o r e m 3.3. For the Hermite operator interpolation problem (3.1) to be solvable, it is necessary and suj~cient that condition (3.16) hold, where

z = I I c ~ c ~ - ~ I tT ~, ~ : 1 , . . , r, (3.1s)

is a linearly independent system of eigenvectors of the matriz T corresponding to the zero eigenvalue.

By analogy with Theorem 3.2, we prove the following theorem.

T h e o r e m 3.4. Let condition (3.16) hold, where Z is defined by relation (3.18), and let T - be a generalized matrix inverse to the matrix T. Then in the case of Qn(x) running through the set (3.17), formula (3.6) describes the whole set of Hermite interpolational operator polynomials of degree n for F satisfying conditions (3.1).

For the most part, the proof repeats the proof of the corresponding Theorem 3.2. Thus, Theorems 3.3 and 3.4 completely solve the Hermite operator interpolation problem (3.1) in the

Hilbert space X. The approach to the constriction of the Hermite interpolants presented above can also be used in the case where the high-order Gs differentials are involved in the interpolational conditions.

Taking into account the discussions and transformations in the course of proving Theorems 3.1-3.4, we formulate a general statement of the Hermite operator interpolation problem and present its solution in the Hilbert space X.

. , . m Let the operator F be ki, i = 1 , . . m, times differentiable by Ggteaux at the nodes {x,}i=l C X. It is required to find the operator polynomial Pn 6 IIn satisfying the following Hermite interpolational conditions:

' ' = F (Xl )ViVi__ 1 ' ' V ~ ,

i = O , . . . , k t , l = 1 , . . . ,m .

(3.10)

Let us introduce the following notation:

12

gn(~,v) = ~ ( u , v ) : , p = l

u, v E X;

T is a symmetric matrix of dimension (k x k),

m

k = ~ ( k , + 1), i----1

r = t t~ m Iltt~l I is a matrix block of dimension (kt + 1) x (ks + 1): l , s = l ,

2471

tl; =

"",,it',,= , j=o ..... k., lltl; I/_-o ..... ~,

o'" ( ~ ' ~ ; )o , ..... o,--~,, ~,=o ..... 0~1 . " OaiOl31 . . .Ofl i g'~ zt + apVp, z~ + 13pv p = l p = l

T - is a generalized matrix inverse to the matrix T, .4 = E - T - T is an idempotent matrix,

{ }m -~ (0 I I . . . v l ~ , i . . . . , k l = F (xt)vivi_ ~ O, l = l '

~.~ = , ( x , ) v , v , _ , . . . v ' , , i=o,.. . ,k, ,=.

kt k2 krn

6o(x) is a function defined by the relation

60(x) = 0 , xT~0; 60(0)= 1; k

i - - - - I

Let us present the two main theorems of this section, the proofs of which repeat, for the most part, those of the corresponding Theorems 3.3 and 3.4.

T h e o r e m 3.5. Let the rank of the matrix T be equal to k - r, r > O, and let

z l c ~ - ~ r l l T

where c'i, i = 1, . . . ,r, are linearly independent eigenvectors of the matrix T corresponding to the zero eigenvalue. Then for the Hermite interpolation problem with interpolational conditions (3.19) to be solvable, it is necessary and sufficient that the condition

E - ~ Z F ~ = e (3.20) zg~ (zg~) ' yT

60 (llZ~'~ll) + IlZa'~ll 2 ]

hold, II'll is a norm generated by scalar product in Rr, and E is the identity matrix of dimension (r x r).

Consider an operator polynomial of degree n of the form

/ f 0 i

= Q'n(X)q- \ ~ 9 - ~ n g ' T - ~ 0o~1 : :'.Oog i P~(x)

where Qn E II~ and

n~ = {e.(~)

gn X Xl-~-~-~ OtpYp, X , i = 0 , . . . , kt , \ p = l 1=1

(3.21)

T h e o r e m 3.6. Let condition (3.20) hold. Then in the case of Q,(x) running through the set (3.22), formula (3.21) describes the whole set of the Hermite interpolational operator polynomials for F of degree n satisfying interpolational conditions (3.19).

Thus, the following results are obtained. The necessary and sufficient condition for the solvability of the Hermite operator interpolation problem with interpolational conditions (3.19) is given. Here the matrix

2472

acting on the vector ~ ~ depends only on linearly independent eigenvectors of the matrix T corresponding to the zero eigenvalue. For a fixed matrix T - , formula (3.21) describes the whole set of Hermite interpolants in the Hilbert space X that satisfy interpolational conditions (3.19). But since the matrix T - is not unique, the expression (3.21) determines the class of Hermite interpolational operator formulas. Note that the Hermite interpolational operator formula from [46] with the Moore-Penrose matrix T - chosen as generalized matrix T + is a particular case of formula (3.21).

By analogy with the results of w of Chapter 1, from the set of Hermite operator interpolants we extract a subset of c-interpolants.


T h e o r e m 3.7. Let condition (3.20) hold, and let II, = II~(F) be a set of c-polynomials in the set (3.22). Then formulas (3.21)-(3.22) describe the whole set of Hermite c-interpoIants satisfying interpoIational conditions (3.19).

The proof repeats, for the most part, the proof of the corresponding Theorem 7.1 from Chapter 1 for operator interpolants.

Coro l l a ry . Let us choose, from the set (3.22), the polynomial Q,(x) constructed ffom the polynomial P,(x) of the best approximation to F in the Hilbert space L2(X) [15]. Then, substituting it in (3.21), we obtain the Hermite interpolant preserving a polynomial of the corresponding degree, the structure of which does not contain the Gdteaux differentials except for those involved in the interpolational conditions.

Let us describe an extremal property of some Hermite operator polynomial of n th degree in the Hilbert space H(0) with scalar product (2.3) from Chapter 1. Denote by II~(F) a set of Hermite interpolational operator polynomials in the separable Hilbert space X that satisfy the interpolational conditions

(i) z t P~ (Bxt)BviBvi_ 1 Bv[ (i) t l . . (3.23) �9 .. = F (Bxl)BviBvi_ 1 .. Bv~, i = O, . . . ,k l , 1 = 1,.. ,rn,

where B is the conditional kernel operator of measure p on X such that ker B = O. Consider the Hermite interpolational operator polynomial satisfying conditions (3.23) of the form

P2(x) = Q,~(x,F) + ao + < ~ - ~ n ~ - aoe~,

Oi gI (xz T+ {OOtl . . . O a i n

where I is the identity operator,

+ apvv, x , i = O, . . . ,k t , p = l ot I = . . - ~ c r , :=O / = 1

(3.24)

n

( u , , ) = p--.1

u, v E X,

T is a symmetric matrix, T ,~ m = IIt IIt,s=l, Ilttstl is a matrix block of dimension (kl + 1) x (ks + 1):

tjt"JJ = Htl;/t,__o,...,k,,j=o,...,k,

• = ~ B , ,:?a~ ---0o~i031 ..-0~3~ g'~ xt + apvv, :c~ + ~3pvv

p = l p = l ct ~ = - - . = a i = f i x = . . . . f l j = 0

T + is the Moore-Penrose matrix pseudoinverse to the matrix T,

~ F (Bz~) O, ~l , B v i B v i - 1 ' 1

2473

O: 0 ~ -

} 11t

~n9 (i) I I . . .Bye , ... k, "= Qn (Bx t )Bv iBv i - i i = O, , I=l'

~'~ = (1, 0 , . . . , 0 , 1, 0 , . . . , 0 , . . . , 1, 0 , . . . , 0 ) ,

ka k2 km

i + (T+~'9, ~9) '

II'A~ ~ - O . ( e , F ) ,

II.ao~'~ll ~ # o, Ao = E - T+T, (3.25)

IIAo611 ~ r o,

Q , ( z , F ) is a polynomial of the best approximation to F in the metric of the space H(0). By direct verification, it is possible to make sure of the fact that the operator polynomial P~(x) is the Hermite polynomial with interpolationM conditions (3.23).


T h e o r e m 3.8. The Hermite operator polynomial (3.24) satisfying interpolational conditions (3.23) is the solution of the eztremum problem

D t1F-5711~(o)- inf IIF-P.IIH(o)- P.EHff(F)

Proof. In order to avoid a bulky presentation and, at the same time, to preserve the generality of discus- sion, let us carry out the proof of the theorem for kt = 1, l = 1 , . . . , m , in formulas (3.23)-(3.25). This corresponds to the case where the Hermite interpolational conditions (3.23) contain values of both operator and polynomial at nodes, as well as the first Gs differentials at these nodes along given directions. Further we will reason as follows. IIn~(F) is a convex set. By virtue of the results of [116], it is a closed set and, since H(0) is a Hilbert space, the set is complete in it. Then the solution of the extremum problem exists and is unique [35]. Further, for P~ e II~ (F) to be an element of the best approximation to F e H(0) in II~ (F), it is necessary and sufficient that the condition

(3.26) ( F - e ~ , e . - P; ).(0) <- 0

hold [35] for 311 P , E H~(F). Put P , ( x ) - P2(x) = R, (x ) e II~(F). Then

(F - P ~ , R , ) u(o) = ( P - Q,, - O~o - < T ~ - ~ , ,~ - aoff~,

{o, T + ~aig~(x, + a v l , x ) , i = 0,1

= (F - Q,,, R,)H(o) - (e~o, R,,)H(O)

g~.(~, + ~ , x ) , i = 0,1 , R~ l = l

}m> "").,0,

H(0) (3.27)

The first addend on the r ight-hand side of equality (3.27) is equal to zero

( F - Q., R.) . (0) = 0,

since Qn(x) is a polynomial of the best approximation to F on the set H , in the metric of the space H(0) and R , ( x ) C II~(F) C II,,. Denote by as E Y, s = 1 , . . . ,2m, the sth coordinate of the vector

T+ ( ~ - ~ r t ~ - OLO~)E Y 2m.

2474

Then we rewrite the third addend on the right-hand side of (3.27) in the form

oi m

• ) ) = a2k-1 (xk,x)~, Rn(x) + a2k E (Xk + o~v~,x)P, , Rn(x) k = l p = l H ( 0 ) k = l -- p = l H ( 0 )

(3.28)

Further, taking into account Lemma 2.1 from Chapter 1, we have

E a2k-I (xk,X)Px, Rn(X) k=l p=l H(o)

= E "'" H (Xk,Vi)x(a2k-1, Lp(vl,v2,. . . ,vp))y t , (evp)#(dvp_,) . . .#(dv,) k-----1 p = l i=1

m 3'l

= E E (a2k-,, L , ( B x k , B x k , . . . ,Bxk))y k = l p = l

rn m

= E (a2k-,, R n ( B x k ) - R n ( O ) ) y = - E (a2k-,, Rn(O))y, k = l k = l

(3.29)

since Rn(Bxk) = O, k = 1 , . . . ,m . Here and later L p ( x , x , . . . , x ) is the pth operator degree of the

P polynomial Rn (x).

Now let us transform the second addend in (3.28) to the form

E a2k E (Xk q'-ctvkl, , Rn(X) = a2k (Xk -t-olvkl,x) p , k = l p = l H ( 0 ) k = l p = l

: E E p a2k(xk'x)Px-l(v~'x) x, Lp(x) k = l p = l H ( 0 )

mn/x..f(P 0 " ( ' ) p-1 �9 a2k~ Xk~ ,_.. cqvi = E E l . , , , OOl...o.,

k = l p = l i=1 x

m n

k = l p = l

k = l p = l

a ! k----I p----1

Lp(x)) H(O)

X V~, Ei=l ~ i U i 'Lp(uI'V2'""UP) y "(dUp)~(dUp-1)'''"(dU1)

"'' ~ (a2k(p--1)'[(v~,vl)~(xk,v2)x.. .(xk,Vp)~

J- (Xk,Vl)x(V~,V2)x(Xk,'a)x'''(Xk,Up)x + "'"

) + (Xk,Vl)x(Xk,V2)x'''(Xk,Vp-1)x(V~,Vp)x],Lp(Vl,V2,...,Vp) Y

X #(dvp)#(dvp-1)" '#(dvl)

[Lp(Bv~,Bzk,. . . ,Bxk ) + L , (Bxk ,Bv~, . . . ,Bzk ) + . . . + L, (Bxk, . . . ,Bxk,Bv~)] )y

k = l p = l Y

2475

m

= R.(Bxk)Bv,)~ ' = o, k = l

(3.30)

since R ~ ( B x k )Bv~ = e e Y , k = 1,. .. , m . Taking into account (3.27)-(3.30), we find

( F - P 2 , R , ) . ( o )

m

k = l m

= -(a0,R.(O))y + ~ (a2k-,,R.(O))y = k = l

- - a0, R n ( e ) ) ) u

(3.31)

Consider the following cases. (1) Let ,zig = ~. Then taking into account the fact that g3 = (1, 0, 1 , 0 , . . . , 1,0) E R2rn and that T + is a self-adjoint matrix, we get

m

E a 2 k _ - - o t o

k = l

Taking into consideration formula (3.25) with .A0g3 = O, from (3.32) we have ( 3 . 3 2 )

m

E a2k-1 -- aO = 0 E ]i. k = l

(3.33)

On the basis of (3.31) and (3.33), we arrive at the conclusion that relation (3.26) holds. (2) Now let r ~ 6. Then, taking into account (3.5) and the equality .A~ = ,40, we find

P . ( e ) = iiA0g ll ,

n , , ( o ) = P . ( e ) - P 2 ( e ) = e e L

meaning that, by virtue of (3.31), relation (3.26) holds again. The theorem is proved. []

In the results presented above, the interpolational conditions at nodes are determined in terms of Gs differentials with repeating directions. Thus, the kth Gs differential at a node contains all k - 1 directions from the (k - 1)th differential at the same node that, generMly speaking, restricts the range of applicability of the Hermite-type operator polynomial. Now we remove this restriction so that directions of the differentiation can be chosen to be arbitrary. Here, as before, the degree of the polynomial n, the number of nodes m and orders hi, i -= 1 , . . . , m , of the Gs differential at nodes are not connected among themselves. With the help of some matrix modification, the necessary and sufficient condition for the solvability of the Hermite operator interpolation problem in a Hilbert space is represented in a more compact form in comparison with (3.20). Under this condition, we describe the whole set of Hermite-type interpolational polynomials as well as the set of interpolants having the property of preserving operator polynomial of the same degree.

Let X be a Hilbert space and Y be a vector space. Let an operator F: X --+ Y, its values at nodes xi E X and Gs differentials of orders up to ki along directions - (1) _ (2) _ (2). _ (k~) _ (k~) ~ ; U i l , u i 2 , . . . , v i i , "ui2 , . . . ,

v(k,) ik, e X , i 1,. , m : F ( x i ) , F ' ( x i " (') F" (x i " (2)(2) F (k')" " (k')v(k') . . .v?a '), i 1, be : " " ) V i l ' ) V i 2 V i l ' " " " ' ' ( X i ) Y i k i i , k ; - - 1 ~ . . . , m , given. On the set II~ of operator polynomials of the nth degree P,~: X --+ Y,

I I , = { P n ( x ) = Lo + L l x + ' " + L , x " } (3.34)

2476

here

P

is a p- l inear con t inuous opera to r ) , it is necessary to find the po lynomia l P,~(x) sat isfying the Hermi te in te rpo la t iona l o p e r a t o r condi t ions

P.(~,) = F ( ~ ) , ' (1)

= F (Xi)Yi l ,

p,,,,, , (2) (2) ,- , , , , , (2) (2) ( 3 . 3 5 ) (Xi)Vi2 Vil : .U ~Xi)Vi2 Vil

= ) - - �9 "-- " ' ' ' O i l " "Ui,ki--1 " ' ' ' U i l '

i = 1 , . . . , m ,

if the p r o b l e m of cons t ruc t ing the po lynomia l satisfying these condi t ions is solvable, and , also, to find the condi t ion for the solvabil i ty of this problem. Under this condi t ion, it is necessary to descr ibe the whole set of Hermi te o p e r a t o r po lynomia l s sat isfying in terpola t ional condi t ions (3.35) and its subse t of in te rpo lan ts having the p r o p e r t y of preserv ing po lynomia l s of the same degree.

Cons ide r the nodes x i , i --- 1 , . . . , L, defined as follows:

^ ,(1) ^ . (2) ~'1 = X l , X2 = Xl -'~ U l U l l , X3 = X l + t Z l U l l ,

a - ( 2 ) _ . ( 2 ) - .(2) _ . (3) :~4 ~-~ X l -1 t- 2 v 1 2 , ;~5 : X l "t- t X l V l l "1- ~ 2 v 1 2 ~ X6 ~--- X l -~- t Z l U l l ,

X7 = X l + a2U(3)12, X8 ~-" X l 3r ~ 3 v 1 3 , X9 = X l + U l U l t + Ot2V

. ( 3 ) ^ . ( 3 ) ^ . ( 3 ) a - ( 3 ) XlO ~ X l -4- U l V l l -4- tX3Ul3 ~ ~'11 ~ X l "JV c t 2 U l 2 3 I- 3 v 1 3 ,

^ .(3) _ v(3) ^ ,(kl) ~ ,(k~) . . a v (k~) e12=:c1+alV~31)+~2,,12 +,.,3 1 3 , . . . , ~ n ~ = z 1 + ~ 1 ~ 1 1 +'~2~12 +" + k, l k ~ , " ' , ^ ~ (k.,) ^ ~ (k..) _ (k,.)'

X'L : Xm "~ t~lUml 2v tX2Um2 "4- ' ' ' "~ Olk,.,, Vrnk,~ ,

where

L1 = 1 + (21 - 1) + . - . + (2 kl - 1) = 2 k '+l - k I - - 1, m

L ~. E ( 2 k i t - 1 _ k i - 1)"

i=l

In t roduce the fol lowing no ta t ion :

p L

[ ' P = (~ ' i '~ ' J )x i , j=l ' p = O , . . . , n , 0 ~

n

F = ~ F V , p=O

F - is a general ized ma t r ix inverse to the mat r ix F,

3* = (F(e,), F(z~), . . . , F(eL)), L

('1 = ~ aibi , i = l

ai E '~a r, bi E N1, a ~-~ ( a l , a 2 , . . . , a m a x k ; ) .

Let us pose the p rob l em of cons t ruc t ing an opera to r po lynomia l f rom H . sat isfying the in te rpola t iona l condi t ions

P . ( ~ i ) = F(~ i ) , i = 1 , . . . , L , P,, C Hn. (3.36)

2477

T h e o r e m 3.9. For the problem (3.36) to be solvable, it is necessary and sufficient that conditions

zT= (3.37)

hold, where Z is a matrix whose elements of rows are coordinates of linearly independent eigenvectors of the matrix F corresponding to the zero eigenvalue. If condition (3.37) holds, then the formula

, , p = O

Q,~ c IIn, (3.38)

describes the whole set of interpolational operator polynomials satisfying conditions (3.36).

Proof. Sufficiency. Let condition (3.37) hold. Then we show that Pn(x, a) is an interpolational operator polynomial for F at nodes Yzi, i = 1, . . . ,L. Denote by gk E ~ - L the vector whose kth coordinate is equal to 1 and the rest are zeros, A0 = E - F-F . We have

( n / p = O

= Qn(2k)+ ( ~ - ~ , , ( E - Ao)~k} = F ( ~ . k ) - ( A ' o ~ - A o ~ , , Yk}.

(3.39)

First we show that A~)~,, = 6 . Indeed, A0 = E - r - r is an idempotent matrix (A 2 = A0) and hence it is simple. Therefore, according to [32], we have

Ao = X A X -1, (3.40)

where A = d i a g { 1 , 1 , . . . , 1 , ~ , 0 ' 0 ' " " 0 } ' r = r k A 0 ,

r L - r

while columns of the matrix X can be interpreted as L linearly independent right-hand eigenvectors for A0 with eigenvalues 1 and 0. It is not difficult to see that eigenvectors of the matrix F corresponding to the zero eigenvalue are the eigenvectors of the matrix A0 corresponding to the eigenvalue 1. Therefore, r = rk A0 is the number of eigenvectors of the matrix F with eigenvalue equal to zero. For the sake of simplicity, we consider the case of rkFa = 1 and ~i = [3ixo, i = 1,... ,L, t[xoll = 1. Then

-" L , = /3k,flk,...,/3~), k = l , . . . , L . r = ( 1 , '

Let 5*be an eigenvector of the matrix F corresponding to the zero eigenvalue, i.e., F5* = 0, 5" = ( e l , c 2 , �9 �9 �9 , e L ) .

We have c1r + c2~2 + " "+ CLCL = 0 and hence 5"f~ (k) = 0,/3 (k) = (/3~,/3~,...,/3k), k = 0 , . . . , n . Taking into account (3.40), we get

A'oQ,, = ( X ' ) - I A X ' - ~ , , = (X ' ) - 'AX'[Lo~(~ L,xo~ (') + . . . + L,~x~'j ( ')]

since for the eigenvector ~'of the matrix A0 with eigenvalue 1, the equality fi(k) = O, k = 0,. . . ,n, is valid. In the case rkFl = p > 1, the relation A~)]~, = ~ is proved analogously.

When considering equality (3.39), we see that the polynomial Pn(x, a) interpolates the operator F at nodes 2i under the condition

Aof f = ~ ~ ( X ' ) - ' A X ' ~ = -~ r Z ] ~:~ = 6 .

2478

The sufficiency is proved. Necessity. Now let the polynomial operator interpolation problem be solvable, i.e., let there exist a

polynomial Pn(x) E I I , for which f t , = ~ . Acting on these vectors from the left by the matrix A~ and

taking into account A~)~n = ~ , we find

A; ' f f = "~ ~ ( X ' ) - ' A X ' ~ :~ = -~ r Z ~ : -~.

The necessity of condition (3.37) is proved. Now let P ~ be some opera tor polynomial of nth degree from I I , satisfying the interpolational

conditions P~ = F(2i) , x = 1 , . . . ,L. By setting Q,(x) = P~ in formula (3.38), we obtain P , ( x , a ) = pO (x). In other words, any interpolational operator polynomial from I I , satisfying conditions (3.36) belongs to the set (3.23). Theorem 3.9 is proved. []

Now we show that the limit P , ( x , a) as a --+ 0 exists, and under certain conditions it is a Hermite-type operator polynomial satisfying interpolational relations (3.35). Let us introduce the block-diagonal matrix

C = d iag{C1 ,C2 , . . . ,Cm},

where the square block Cj = ( A j ) , j = 1 , . . . , m ,

A s =

1 0 0 0 . . . . . . . . . 0 1 1

- - 0 0 0 . . . . . . 0 o~ 1 OL 1

1 1 I 1 0 0 . . . 0

O/10~2 O~10~ 2 (9/10/2 O~10/2

(3.41)

�9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(1)k'/+l 0 0 . . . . . . . . . . . . 1 O Q O t 2 �9 �9 �9 O ~ k j ~ 1 0 t 2 �9 . . O ~ k j

The rectangular matrix A s is a matrix of the maximal rank of dimension (k s -t- 1) x (2 k j + l - k s - 1), Bj is a rectangular matr ix of dimension (2b+l _ 2kj - 2) x (2b+l - kj - 1), the elements of each row of which are zero except for one element equal to 1. The matrix B s is chosen so that the matr ix C s is an invertible matrix. We can always achieve this since Aj is a rectangular matrix of maximal rank equal to k s + 1. It is clear that then the square mat,'ix C defined by relation (3.41) will be invertible as well. Further, putting

n

g(u, v) = Z ( u , p = 0

we transform the second addend on the right-hand side of (3.38) with the help of the matrix C:

/ n ) p L . ~ - ( C ~ C ~ n , C - ' x ) l L ~ (3.42) " ~ - - ~ n , r-- E {(~' i 'X ' )X}i=l -- ( C ' ) - ' r - C{g(x, i , "i=1/"

p = 0

Now consider the limit in (3.42) as a -+ 0. It is not difficult to see that

{ _}m lira C ~ F( i ) ( x j , (i) (i) _ (i) i = O, ,hi , F ( x j ) , = )vii vj,i_l " " u j l , . . . . . . , F ( z 3 ) ol --I. O , , j : l

{ !}m lim C = Q(i)(xj~ (i) (i) _ (i) i = O, , k j , Q , , ( x i ) , . . . Qn(x j n )Yji •j,i--1 " ' 'V j l ' "'" , a- -+0 "- j = l

lim C ~ g ( 2 i , x ) }ira a'-~'O " OOtl . . . oa ig Xj + E (i) -~- OlpUjp , X p = l = - . --=-- a i -..=-0

i = o , . . . ,kj , ,g(xs, ,g(xs, x!} m j = l

l j

= ] ~ , (3.43)

= ~ . ~ , (3.44)

=y(x), (3.45)

2479

where lj = 2 k~+~ - 2ky - 2, ~ o = 0. Pu t T - ( a ) = ( C ' ) - ~ F - C -~, T(a ) = CFC'. Here T - ( a ) is a generalized inverse matr ix to the

matrix T(a) . Indeed,

T ( a ) T - ( a ) T ( a ) = C F C ' ( C ' ) - ' F - C - ' C P C ' = c r r - r c ' = c r c ' = T (a ) .

Let Ao(a) = E - r - r , . 4 ( ~ ) = E - T - ( a ) T ( a ) . Ao(a) and A ( a ) are idempoten t matrices, while the matrix `4(a) can be represented in the form

`4(a) = E - T - ' ( a ) T ( o ) = E - (c')-'r-c-'crc' (E - A o ( ~ ) ) c ' (3.46)

= E - ( c ' ) - ' r - r c ' = E - (C')- ' - (C')- ' Ao(a)C'. Now we show that matrices T(a ) , T - ( a ) , and `4(a) have limits as c~ --+ 0. To do this, it is sufficient

to show that there exists the limit Iim~,-~0 T(a) = T, since from the relations

T ( a ) T - ( a ) T ( a ) = T(a) , .4(a) = E - T - ( a ) T ( a ) (3.47)

the existence of the limits lira T - ( a ) = T - and l im .4 (a ) = . 4 a - + 0 a -+0

follows. Further , it is possible to show that the limit

lira C (a )F (a )C ' (a ) = T 0r

exists and that the matr ix T can be represented in the form m

g ( x t , x s ) . . .

O (x, . . .

2 + g . . .

p = l / ' ] 0

Og(Xt' Xs "3V MlUsl )0 ^ .(1) (1)

o~g(~, + ~ , ~ . , ~ + ~lV,, )o

03g(xt_~t_Ep____l ,2) _ ) t _ f l l V s l ) 0 2 OtpVtp , Xs (1)

. . . . . . . . . . . k ; . . . . . "~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oqktg(xt Jr- ~ O~pl)}pt),xs) . . . aktWlg(xt Jr- Ekpt-_l otpV?pt),xs Jr" f l l V ? l ) ~

\ p----1 / 0 \ -- / 0

g(xt , Xs) �9 . . ~,g~zt, x~ + . 1 ~ 1 Jo

g(x t , x s ) "'" 'JgkXt'Xs q-PlUs1 )0

Ok'g (x t ,Xs -~- Ek~l/~p?)!k'))O . . .

v O) x k. a v(kO)'~ ok'+lg(xtTa~ tl ' s -FEp=I~ 'P sp .]0 "'"

ppVsp )0 " " "

g(x,,~,)

=, =.)o, . . . . . oLu : ~ z : ~ ~(z;; ~ , 4 ~ ; 2 ~ i ; . ) \ . . . . . . . . . : ; , ? , 4 ; , . . . . . . i ; : ) . . . . ~

Here

( s ) c~ v (r) O"+qg x, +/___., p tp , x , + fipv~) p = l p = l 0

(Xt 21- s !r .~_ ' c~, . . . . . c~----fl: p = l p = l

Thus, on the basis of (3.47) the existence of matrices .4 and T - follows.

. . . . . flq=O

(3.48)

2480

T h e o r e m 3.10. For the Hermite interpolational problem (3.35) to be solvable, it is necessary and sufficient that the condition

Z~:~ = -~ (3.49)

hold, where Z is a matrix, the elements of rows of which are coordinates of linearly independent eigenvectors of the matrix T corresponding to the zero eigenvalue. I f condition (3.49) holds, then in the case of Qn running through II~ the formula

(3.50)

describes the whole set of Hermite-type operator polynomials of the nth degree satisfying interpolational conditions (3.35).

Proof. Sufficiency. Let condition (3.49) hold. Let us show that in this case polynomial (3.50) is the Hermite-type polynomial for F satisfying conditions (3.35), i.e., let us show that the Hermite interpolational problem (3.35) is solvable. Let e'k be a vector from RL, whose kth coordinate is equal to 1 and the rest of which are zeros. We have

tXs)Vsk Vs,k--1 "'" a l - - "

+ - r - o 9 x, + , , , ) p - - - - 1 0

- - ~ n t s) sk Vsik--1 " ' 'V s l "4" -- --

---- F(k)'xt sl'~v(k)v(k)sk s ,k - I . . . . v~kl , "3t" ( . 4 ' 0 9 . 4 ' ~ n 9 , ~X:>"

(3.51)

v (k)~ In Here k is the number of the column of the matrix T coinciding with the vector ak~(x~ + ~'~k=l p sp ]0"

the course of proving Theorem 3.9, it was shown that A ~ n = -~. From this it follows that

A'o ]~,, = -~ ~ C A'o C - ' C ]~, = --~. (3.52)

Passing to the limit in (3.52) as a --+ 0 and taking into account (3.44), (3.46), we get

`4"~'r,9 = "~. (3.53)

Taking into consideration (3.53), from equality (3.51) we conclude that for P~(x) to be the Hermite polynomial and hence for the interpolation problem to be solvable, it is sufficient that the condition

.4'b:~o = -~ (3.54)

hold. ,4 is an idempotent and hence a simple matrix. Therefore, it can be represented in the form .,4 = X A X -1 , where X is a matrix, the columns of which are linearly independent eigenvectors of the matrix .4, and A is a diagonal matrix of its eigenvalues consisting of units and zeros. Since the eigenvectors of the matrix T corresponding to the zero eigenvalue are the eigenvectors of the matrix .4 with eigenvalue 1, reasoning in same manner as in the proof of Theorem 3.9, we get

(3.55)

Condition (3.49) holds, meaning that condition (3.54) holds as well. Then from (3.51) and (3.53) the solvability of the Hermite operator interpolation problem (3.35) follows. The sufficiency is proved.

2481

Necessity. Now let the problem (3.35) be solvable. This means that there exists a polynomial P, (x ) E

II , such that -fl-9 = ]~'~. Taking into consideration (3.53) and (3.55), we get A'i~=~9 = A']5~,9 = ~ r

Now let P~ be some operator polynomial of the nth degree from II , satisfying interpolational conditions (3.35). By setting Q, (x ) = P~ in formula (3.50), we have P, (x ) = P~ Thus, any interpolational operator polynomial from I-l, satisfying conditions (3.35) belongs to the set (3.50). Theorem 3.10 is proved. []

Denote by 1-I, (F) the set of operator polynomials Pn(x, F) of the nth degree that are operator functions of F. We call an operator polynomial P,~(x, F) �9 1-In(F) a c-polynomial if it satisfies the condition

P,~(x,F) = F(x) for all F e I I . , x �9 X.

Within the framework of Theorem 3.10, by analogy with the corresponding theorem from [46], the following result is valid.

T h e o r e m 3.11. Let condition (3.49) hold and let Hn = 1-I~(F) be the whole set of operator c-polynomials. Then in the case where Qn(x) runs through H~(F), formula (3.50) describes the whole set _f Hermite-type polynomials of the nth degree preserving operator polynomials of the same degree.

Remark. Let P*(x, F) be some c-polynomial. In [47], the whole set of c-polynomials of the n th degree is shown to be described by the formula

P,~(x,F) = P,~(x,F) - Pn(x ,P*) + P*(x ,F) for all P,~(x,F) �9 1],(F).

Now we show that within the framework of equivalent transformations the main formulas of the Hermite operator interpolation can be reduced to formulas containing vectors and matrices of lower dimension. This concerns both the conditions for solvability of the interpolation problem and interpolational formulas.

We rewrite the Hermite-type operator polynomial (3.50) in the form

where

vectors

L I

O. e 1-I., (.) = ~ ~ib~, ~ e Y, bi E R1, i=1

-~ (1 ) = ~F(i)tx .~v(i)v(i) _ (i) i = O, )rrt [ ' 3 ' j i j , i - l ' ' ' V j l ' . . . , k j , f ( x j ) , F ( x j ) , . . . , F ( x j ) j = l '

Y

lj

) ,.~(i)t x ,v(i) (i) .(i) i = 0 , k T,Q,~(xj) ,Q.(z j ) , Q . ( z j i=~ --~ t~n ~ J) j i Vj, i--1 " ' ' V j l ~ " ' ' ~ �9 " ' ' ~ ~ r

lj

{o, ( )I ._ v ' , . (i) f f~ (x )= Oa~ Oaig xj + ?__. c~,vip, x

p = l c ~ l = " ' = ~ i = O

i = 0 , . . . ,k j , g ( x j , x ) , g ( x j , x ) , . . . , g ( x j , x ) , ~" ~ " ) j = l

~'(1)(x), ~ ! l ) ( x ) , and ffl(x) are of dimension

?n

L1 = Z (2k~+l - kj - 1), lj = 2 k~+l - 2kj - 2, j = l

0 n

=0, 1 p=O

2482

7'1- is a generalized mat r ix inverse to the matr ix 7'1 = IIT,,Ih%=l of dimension (L3 x L,) , while the matr ix block Tt~ is defined by formula (3.48).

Let gi = (c~, 4 , c i ) i = 1, be linearly independent eigenvectors of the matr ix TI corre- " ' ' ~ L1 ' " ' ' ' r t l '

spond ing to the zero eigenvalue, i.e., Tl~i = O. Since T1 is a real, s ymmet r i c and hence s imple ma t r i x [32], we have n l = L~ - r, where r = rk T1. Let us const ruct the mat r ix Z1 = lie',, ~'2,..., ~',, II T, the e lements of rows of which are coord ina tes of vectors c'i, i = 1 , . . . , n l .

Let

~:~(2) f F ( i ) r x .~v(i)v(i) _ (i) i = 0 , . . . k j } m = [ ~ 3J ji j , i -1 " " v i i , ' j - l '

~(n2 { } m , (i) (i) .(i) i = 0 , . . . k j , ) = Q( i ) (xJ)YJ i Vj'i--1 " ' " V J l ' ' j = 3 '

�9 ( i )

ff2(x) = 0 o q . ' . 0 o q g x j q - a , v j p , x , i = O , . . . , k j , , p : l j = l

m and let T2 = IIT-II.=~ be the ma t r ix of d imens ion (L2 x L2), where

m

L2 = ~ ki -4- m, i = 1

T t $

�9 f 3 . ( 1 ) ~ g ( X t ' X s ) "" O g ( x t ' x s 2 t - ' l t l s 3 ) 0

. (3) ~ . (1) f~ . (3)'~ O g ( x t -'~ ~ l V t l ' X s ) o " ' " Oq2g(xt nt-ct lVt3 ' X s -31- ~ l O s l )0

2 z '" ) ( ' o - ' " ' a p V t p , X , " ' " Oag X t + Y'~p=3OtpVtp ' m s ~ ) 0

p = l 0

. . . . . . . . . . "~; . . . . . k . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . ~ 1 7 6 kt ( t ) k t + 3 kt (kt) (3) o g(,,, + o,,,vtp . . . o g(x, + +,,v,1 ) p--1 O 0

. . . a v(k,)~ Ok" g (Xt, Xs -[- E k ~ 31~'p sp ,]0

_ (3) k, N v(k~ �9 "" o k ' + l g X t -~-~ l"Ut 1 , X s - 4 - E p = l t ~ p sp ] 0

Ok'+~g( zt + ~ = 1 (2) k. ~ (k.)~ ~vvtp , xs + ~_,v=l . . . . p v , p ) o

. . . . . o;. ; i , - ?~ 7 ;z~; ~ fi ; ; .... : ; z ~ ; ; i ~ ; ;~ " ' " Y ~ t"l- 2~p=1 P tp ,;:gs-I- ~..~p-~3PpVsp ) 0

(3.5r)

F r o m the s t r u c t u r e of mat r ices T1 and T2, it follows tha t rk T3 = rk T2 = r. T2 is a s imple matr ix , therefore it has n2 = L2 - r l inearly i ndependen t eigenvectors (~, i = 1 , . . . , n2, co r respond ing to the zero eigenvalue, i.e.,

, .. d i T 2 ~ = 0 , ( ~ = (d~ d l , . , L=), i = 1 , . . . , n 2 .

Consider the m a t r i x Z2 = 1 1 ~ , ~ , . - - , d ~ = l l T and the condi t ion

z~Y (~) = ~ . (3.5s)

T h e o r e m 3 .12 . C o n d i t i o n s (3.49) and (3.58) are equivalent .

Proof. Let Z l ~ ( 1 ) = ~ . For the sake of simplicity, we consider the case where the ma t r i ces T1 and T2 are cons t ruc ted f rom one node. S t ruc tu ra l ly these matr ices (see formulas (3.48) and (3.57)) can be wr i t t en in the fo rm

2483

=

al l a12 . . . a lk al l . . . a l l a21 a22 . . . a2k a21 . . . a21 . . , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a k l ak2 . . . a kk ak l . . . a k l

a l l a 1 2 . . . a l k a l l - - . a l l

a l l a 1 2 - . . a l k a l l . . . a l l

and

a l l a 1 2 �9 �9 �9 a l k

7"2 = a 2 1 a 2 2 . . . a2k

a k l ak2 �9 �9 �9 akk

Let c'i = ( c~ ,c i2 , . . . , c ~ , ) , i 1 , . . . , n l , be linearly i ndependen t eigenvectors of the ma t r i x T1 correspond ing to the zero eigenvalue,

T1 ~i = 0, i = 1 , . . . , n~, n , = L1 - r. (3.59)

W i t h o u t loss of generali ty, the minor of Order (r • r) in the uppe r left corner of the m a t r i x 7"1 can be assumed to be nonzero. T h e n we represent the sys tem (3.59) wi th respect to coord ina tes c~, i = 1 , . . . , L1, in the fo rm

r L 2 L1

a k j c } = - E a k j c } - - a k l E C}, k = 1 , . . . , r , i = 1 , . . . , L 1 , (3.60) j----1 j = r + l j=L2-1-1

�9 " i where c], c~, , c r are chosen to be basic variables. We set the last L1 - L2 variables c i c i c i " ' " L 2 - t - 1 ' L 2 - 1 - 2 ' ' ' ' ' L I

equal to zero. T h e n sys tem (3.60) acquires the form

r L 2 E i - - - - - - a k j c j akjC~., k =- 1 , . . . , r. (3.61) j = l j=rq-1

By solving this sys tem, we f ind L2 - r = n2 linearly independen t eigenvectors ~ = (Qi, c2i,. . . , ciL2) such that

T2(~ = 13, i = 1 , . . . ,n2. (3.62)

Thus , vectors ~ , i = 1 , . . . , n2, axe l inearly independen t eigenvalues of the ma t r i x T2 co r respond ing to the zero eigenvalue.

Now consider the mat r ices T1 and T2 cons t ruc ted f rom m nodes. Let condi t ion (3.59) hold, where

r = rk 7"1. Let us find vectors ~ , i = 1 , . . . , n2, satisfying ( 3 . 6 2 ) . Since the ma t r ix 7"1 consists of blocks (see formula (3.48)), we see tha t the last l j , j = 1 , . . . , m , elements in each row of the block are equal to the first e lements of this row. Then , ju s t as for the ma t r ix T1 cons t ruc ted f rom one node, we wri te Eq. (3.59) in the coord ina te form, choosing for basic variables such variables whose coefficients in the sys tem of equat ions are not equal to the last l j , j = 1 , . . . , m , elements of rows. T h e n we proceed in the same m a n n e r as in the

case of mat r ices 7'1 and T2 cons t ruc ted f rom one node. We get vectors ~ = (Cl ,C2, . . . ,CL2) , Z _ _ i i i " -- 1 , . . . , n 2 ,

such t ha t T2~ = 6, i = 1 , . . . , n2. Then , taking into account the s t ruc tures of vectors -ff~(1) and 0 (2), we

have z , = -6 = - 6

Now let us prove t ha t Z 2 ~ ~'(2) = -6 =~ Z ~ I) = -6. As before, for the sake of s impl ic i ty we consider

the mat r ices 7"1 and T2 cons t ruc t ed f rom one node. Let o~ = ( c~ , c~ , . . . , c~2 ) , i = 1 , . . . ,n2, be linearly independen t eigenvalues of the ma t r ix T2 cor responding to the zero eigenvalue. Let us cons t ruc t linearly i ndependen t vectors c'i, Tlffi = {), i = 1 , . . . , h i = 1 , . . . ,n2 -1- L1 - L2. Let us in t roduce the vectors

~i ~- C , C 2 , . . . ~ C L 2 , �9 . . . , , , . . . , n 2 .

Lx~-L2

2484

They are l inearly independen t , and T15"i = 0. The matr ix T1 has nl = n2 + L1 - L2 l inearly independent eigenvectors corresponding to the zero eigenvalue; therefore, L1 - L2 more of such vectors should be found. Suppose tha t among all c'i, i = 1 , . . . ,n2, there exists at least one vector 5'k, 1 < k < n2, wi th nonzero first coordinate. Then we take the vector ~), and interchange its first coordinate wi th zero coordinates. Since c'k has L1 - L2 zero coordinates , we obta in L1 - L2 more linearly independent vectors of the form

4,0, ,0) = , ' ' ' , L 2 , " ' " ,

: (0,4, ,4, , 0,4, ,0),

1 = ' ' " ' L 2 ' O , O , . . . , C �9

It is not difficult to see tha t all vectors c-i, i = 1 , . . . , n l , are linearly independent and TlC'i = (), i = 1 , . . . , n l . Consider the case where all the vectors c~, i = 1 , . . . , n2, have first coordinates equal to zero. Then we

take, for example, the vector 5"1 and construct the following system of vectors:

c'n2+, : (0, c 1 , . . . , c ~ 2 , 1 , - 1 , 0 , . . . , 0 , 0 ) ,

= .,0,0),

= ( 0 , 4 ,

= �9 , 0 ,

It is not difficult to show tha t fi'i, i = 1 , . . . , n l , are linearly independent vectors and Tl~i = O. Now consider matr ices 7'1 and 7'2 constructed from m nodes. Let T2di = 0, i = 1 , . . . , n2. Let us find

v e c t o r s ~i = (C~, ci2, . C' -~ "~ �9 " , LI), i = 1 , . . . , n l , such that Tlci = 0 , w h e r e n l = n 2 + L 1 - L 2 . As was noted above, each row of a block of the mat r ix T1 contains elements equal to the first e lement of this row. We set the coordinates of vectors c'i, i = 1 , . . . ,n2, corresponding to these elements equal to zero. We set the remaining L2 coordinates equal to the coordinates of vectors ~ . We get n2 l inearly independent vectors fi'i. To construct the missing L1 - L2 linearly independent vectors, we proceed in the same manner as in the case of m a r r i e s T1 and 7"2 cons t ruc ted from one node. Thus, we find nl l inearly independent vectors 5'i such that T l • i = 0 , i = 1 , . . . , n , . The structure of vectors ~'i, i = 1 , . . . , n l , ~ , i = 1 , . . . ,n2, T (1), and ]~(2) is such tha t Z 2 f f (2) = -~ ~ Z1 ~ ( 1 ) = ~ . Theorem 3.12 is proved. []

T h e o r e m 3 .13 . For the Hermite operator problem (3.35) to be solvable in a Hilbert space, it is necessary and sufficient that conditions (3.58) hold. If these conditions hold, then in the case where Q, runs through I I . the formula

describes the whole set of Hermite operator polynomials of the nth degree. Here T 2 is a generalized matrix inverse to the matrix T2,

Proof. The proof of the theorem repeats, for the most part , the proof of the corresponding Theorem 3.10. Thus, the necessary and sufficient conditions (3.58) for the existence of the Hermi te- type operator

polynomial are obta ined, and under these conditions the whole set (3.63) of such interpolants is described. The vectors and matr ices involved in formulas (3.58), (3.63) have lower d imension than the corresponding vectors and matr ices in (3.49) and (3.50). Taking into account the s t ruc ture of matr ices 7"1 and T2 (see formulas (3.48), (3.57)), the mat r ix 7"2 can be obtained from the matr ix T1 by deleting repeat ing rows and

columns, while the vectors ~:t(2), ~(n2) and ~2(x) can be obtained from the vectors ~=~(1), ~n(1), and ~l(x) by deleting repeat ing coordinates , i.e., values of the operator at nodes. [3

2485

Remark. The condition Z2"ff (2) = ~ ' is equivalent to the condition for the solvability of the Hermite operator problem given in [46] for the case of interpolational conditions with Gs derivatives along repeating directions.

w P o l y n o m i a l i n t e r p o l a t i o n of o p e r a t o r s in a r b i t r a r y vec to r spaces Let X and Y be vector spaces over R1, let F: X ---r Y, and let H, be a set of operator polynomi-

�9 m als P,~: X ~ Y of degree not exceeding n, {x~}i=l C X. The polynomial operator interpolation problem �9 lies in finding a polynomial from H,, satisfying the interpolational conditions

Pn(xi) = F(xi) , i = 1 , . . . , m . (4.1)

Here, in contrast to the classical interpolation of functions, the numbers m and n are not connected with each other.

Before turning to the presentation of the main theorems of this section, we present some auxiliary results needed hereafter. Let there be k linearly independent elements, 1 _~ k _~ rn, among elements

�9 m {x,}i= 1. Without loss of generality, we assume them to be the first k nodes xl, x2 , . . . , xk. Then

k

xi = Z aipXp' c~i v E R1, i = 1 , . . . , rn. (4.2) p = l

The following results are valid.

L e m m a 4.1. In the algebraically adjoint space X* there exist k linear functionals lO')(x), p = 1 , . . . , k , defined everywhere on X such that

l(P)(Xq) = ~pq, p, q = 1, . . . , k,

5pq is the Kronecker symbol.

The statement of the lemma is contained in Proposition 1.4.3 from [89]. Let us construct m tinear functionals

k

i= l , . . . , m , p--1

as well as the following matrices needed hereafter: n

m E F, = [[tT(xj)[[i,i= 1, p= 1 , . . . , n , P = F,, p----1

(4.3)

F - is a generalized inverse matrix to the matrix F, A0 = E - F - F is an idempotent matrix, and E is the identity matrix.

Let us introduce the following notation: m m = {f(xi)}i__l, ~ n - - {P(xi)}i=,,

(s, t): (ym, Rm) -+ Y is an operator form of the type

Trt

( s , t ) = ~ s i t i , si e Y, ti �9 n , , i = l

~ = ( 1 , 1 , . . . , 1 ) e R m ,

5o(X) is a function equal to zero for x ~ 0 and equal to 1 for x = 0, I1"il is a no rm in Rm, Z = t1~'1~'2-- rtl T, where c'i, i = 1, . . . ,r, are linearly independent eigenvectors of the matrix F corresponding to the zero eigenvalue. Denote by I I~ the set of operator polynomials of degree n of the form

II,~ = { P n ( x ) + [5o (Z ' f f , Z~} + 11Z '112 - P,'(O)] [ 1 - 5 o ( [ I Z ~ l l ) ] : P . e II.}. (4.4)

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L e m m a 4.2. For any Pn E IIn the relation

z ~ . = P.(O)z~ (4.5)

is valid.

The proof of this lemma uses the specific nature of the matrix I', which is a Gram matrix for some system of vectors •i, i = 1 , . . . , m, coordinates of which can be expressed in terms of numbers ais, i = 1 , . . . , m, s = l , . . . , k ,

r = = ,~ , .~ . = i(r r m ' J)l i , j=l

p = l i , j=l L s = 1 a i,j=l

On the whole, the proof completely repeats that of Lemma 3.2 from Chapter 1. The following theorem is the main result of this section.

T h e o r e m 4.1. For the polynomial operator interpolation problem to be solvable, it is necessary and sufficient that the condition

( ) E - ~0(tlZ~'ll) + IlZ~'ll 2 Z ~ = • (4.6)

hold, where E is the identity matrix of dimension (r x r). If this condition holds, then in the case where Qn(x) runs through I I~ the formula

p I ( x ) = Q . ( x ) + - ~ n , r - E { l f ( x ) } , = l p = l

(4.7)

describes the whole set of interpolational polynomials for F of the nth degree in the vector space X satisfying conditions (4.1).

Proof. First let us prove the necessity of condition (4.6). Let P,(x) be an operator polynomial from [In that is interpolational for F at nodes {xi}'~=l C X. From the equality 5i~ = ~5~, and Lemma (4.2), it follows that

Z ~ = Z ~ , = P,(O)Z~. (4.8)

If Z g = 6, then from (4.8) we get Z ~ = 6, and condition (4.6) holds. Now let Z~ ' r 6. Then from (4.8) we find

( z T ~ , z ~ ) = Ilz ll2P.(e),

( z T , z~ )

P.(O)- Ilz ll Taking into account (4.9) and (4.8), we have

Z ~ - ZgPn(O) = Z ~ (Z~,Z~) Zf = Z ~

and condition (4.6) holds again. The necessity is proved.

(4.9)

ze(zr - ~ = (E zr162

Let us prove the sufficiency. Let condition (4.6) hold. Let us show that polynomial (4.7) is an interpolational polynomial for any Qn (x) E rl~ (F) and hence the polynomial operator interpolation problem is solvable. We have

P . ( x k ) = Q . ( x k ) + T - - ~ . , F - X ~ l~(xk) /=, = O n ( x k ) + T--~,~, r-I '~ p = l

:

(4.10)

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Here e'k E R,n is a vector, whose kth component is equal to 1 and the rest o f which are zeros ,

Ao = X A X -1,

where

(4.11)

x = Ile, smtl,

0,0,...,0},

c-'/, i = 1 , . . . , m, are linearly independent right-hand eigenvectors of the matrix Ao, among them c'1fi'2--" c'r being eigenvectors of the matrix 1-' corresponding to the zero eigenvalue, rk 1-' = m - r, 1 _< r _< m. Taking into account (4.11), from (4.10) we find

Vn(Xk) = F(Xk)- ( x t - l (AX t~_ AXt~n), ~k I

where I T.

= I m - - r

Consider the following cases. (1) Let ZoF= 6 ~:~ Zg= 6. Then from (4.6) it follows that Z ] ~:t = ]~ ,~

Z0 ~:t = ]~ and according to (4.12) the operator polynomial Pn(x) is interpolational for F at nodes xk, k = 1 , . . . ,m. (2) Let Z0~' # 6 ~, Zg # 6. But then, taking into account the fact that Q,,(x) E n~ we h a v e

<zYt'z ) z0 = -6 tlz lp

if condition (4.6) holds and Pn(z) is an interpolational polynomial for F at nodes Zk, k = 1 , . . . , m. Thus, the sufficiency of condition (4.6) for the solvability of the polynomial operator interpolation problem is proved.

Since, as was shown earlier, under condition (4.6) the operator polynomial (4.7) is an interpolational polynomial, it remains for us to show that formula (4.7) describes the whole set of operator interpolants for F at nodes Zk, k = 1 , . . . , m, in the vector space X. Indeed, let P~ be some fixed interpolant for F. Then, taking into account the fact that P~ E n ~ we set Q,(x) = P~ in (4.7) and immediately obtain P~(x) = P~ Thereby we show that P~ belongs to the set of polynomials described by formula (4.7), and hence this set contains all operator interpolants for F at nodes xk, k = 1 , . . . ,m, of the n th degree. The theorem is proved. []

Note that, in view of the nonuniqueness of the generalized inverse matrix F - , formula (4.7) describes a class of interpolational operator formulas in the vector space X. The formula in [46] is a representative of this class in the case where the Moore-Penrose matrix F - is chosen as the matrix F +.

T h e o r e m 4.2. The polynomial operator interpolational problem (4.1) is always solvable for m < n + 1.

The proof completely repeats the proof of the corresponding Theorem 8.1 in Chapter 1. Just as in the case of classical interpolation, the extraction, from set (4.7), of the subset of operator

polynomials preserving the polynomial of the corresponding degree is of great interest. Denote by II~ the set

I I~ { P , ( x ) + [60 (Z~:t'Z~)

where II~(F) is the set of c-polynomials defined by relation (7.1) in Chapter 1. The following theorem is valid.

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T h e o r e m 4.3. Let condition (4.6) hold. Then in the case where Q,,(x) runs through II~ formula (4.7) �9 m describes the whole set of interpolational operator polynomials of degree n for F at nodes {x,}i= 1 C X in

the vector space X preserving polynomials of the same degree.

The proof of the interpolationality of (4.7) under the conditions of the theorem completely repeats the proof of the sufficiency of condition (4.6) in Theorem 4.1. Further, let F E I'I,, and Qn = H~ Then

= ~],~ and hence P~(x) = Qn(x) = F(x) E IIn, i.e., P~(x) preserves a polynomial of the corresponding degree. Now let Q~ ) E IIOc(F) be some interpolational operator polynomial for F having the property of preserving a polynomial. Then, by setting Q,~(x) = Q~ ) in (4.7), we get PIn(X ) = Q~ i.e., formula (4.7) describes the whole set of interpolational c-polynomials in the case of Qn(x) running through II~ The theorem is proved. []

For the description of the whole set II~(F) of c-polynomials of degree n defined by formula (7.1) in Chapter 1, the existence of at least one c-polynomial is needed. Let us dwell for a while on the question of the existence of an operator c-polynomial. Consider three examples of constructing such polynomials.

1. Let X, Y be Banach spaces, let .7- be the vector space of continuous operators from X to } , and let IIn be the linear manifold of continuous operator polynomials of degree not exceeding n. The following lemmaz answer the question as to the existence of an operator c-polynomial.

L e m m a 4.3. The linear manifold IIn E ~" is closed.

The proof of the lemma employs the result on the limit of a sequence of continuous operator polynomials of degree not exceeding n, which is either a continuous operator polynomial of degree not exceeding n or is zero [116].

On the basis of Lemma 4.3, Theorem 1.1 from [19], and Corollary 2 from [69, p. 475], the following statement is valid.

L e m m a 4.4. Let ~ be a reflexive Banach space. Then for all F E ~ on any bounded convex set from IIn there exists a polynomial of the best approximation for F, which is obviously a c-polynomial.

2. Let X be a vector space and let F(x) be n-times Gs at point x = O. Then the polynomial

F'(O) P.(x) = F(O) + 1! x + . . . + n! " "

is a c-polynomial, where F(k)(O)x k is the Gs differential of the kth order of the operator F. The proof is obvious.

3. Let {x~} be an algebraic basis in the vector space X and let X* contain the family {l~} dual to {x,~}, a E T, i.e.,

l i ( x j ) ~-- 5i j , i , j = T,

5ij is the Kronecker symbol. Then any element x E X can be represented in the form

x = E I i ( x ) x i l E T

with li(x) being nonzero for not more than a finite set of indices from T (in general, this finite set of indices depends on x).

Let us give examples of operator c-polynomials of first and second degree in the vector space X:

F) = F(O) +

P2(x;F) = F(O) + E r + E r x j ) l i (x) l j (x) , l E T i , j E T

2489

where operators ~01:X --+ Y, r --~ Y, and r 2 --+ Y are of the form

~,(u) = F ( u ) - F(O), u E X,

~)1(1~) : 2 F ( u ) - ~ F ( 2 u ) - F(@), u E X,

1 [F(2u + 2 v ) - 2F(u + v ) - F(2u)+ 2 F ( u ) - F(2v) + 2 F ( v ) - F(O)] r v) = , u, v E X .

Let us show that PI(x, F) and P2(x, F) axe operator c-polynomials of the first and second degree. First let

F(x) = Lo + L1 x.

Then

PI(x ,F) = Lo + E [Lo + L,x, - Lo]li(x) iET

"~ L~ -~- E i , x i l i ( x ) : i O -}- L 1 E l i(x)xi = iO "~ i l x -- F(x ) . lET lET

Further, let

Then the following relations are valid:

F(x) = Lo + Llx + L2x 2.

{ I[Lo + L,2x, + L2(2xi) 2] - 3Lo} l i ( z ) E r ---- E 2Lo + 2Llxi + 2L2x 2 -- -~ lET iET

= E n lx i I i (x ) = nl E Ii(x)xi -~ n ix; i E T i E T

r xj)t;(x)Zj(x) i,jET

= ~ ~ Lo + L,(2xi + 2xj) + L~(2~, + 2xj) 2 - 2Lo - 2L,(~, + x,) i , j ~ T

- 2L2(z~ + x j ) 2 - Lo - L12zi - L2(2z i ) ~ + 2Lo + 2Llx~

+ 2L2x~ - Lo - L,2xj - L2(2xj) 2 + 2Lo + 2ilxj + 2L2x~ - Lo}l i (x) l j (x)

i,jET " lET iET

Therefore, for F(x) = Lo + LlX + L2 x2 we have

P2(x,F) = Lo + L ,x + L2x 2 = F(x),

which proves the statement. Now we show that under certain conditions the initial system of nodes generates a broader set of

interpolational nodes, which is the union of the given system and some span. Let {x~} be an algebraic basis in the vector space X that contains nodes xi, i = 1 , . . . , k (xl , x2 , . . . , xk

are linearly independent). The existence of such a basis is guaranteed by Theorem 1.4.5 from [89]. Denote by Z0 the span constructed from the system of elements {x,~}\{xi}ki=l .

2490

�9 m T h e o r e m 4.4. Let X be a locally convez space and let a system of nodes {x,}i=l be such that de tF # 0,

m m

T = { F ( x i + y ) } i = 1, ~ n = { Q n ( x i + y ) } i = 1, y E Z o .

Then for ally E Zo the system of nodes {zi+y}iml iS an interpolational system for operator polynomial (4.7).

Proof. According to Theorem 1.36 from [63] and Corollary 1.7.2 from [89], there exist linear functionals l(P)(x), p = 1 , . . . , k, defined everywhere in X such that

I(P)(xi q- Y) = 5ip, i,p = 1 , . . . , k, for all y E Z0,

5ij is the Kronecker symbol. Obviously,

F - = F -1, A0 = E - F - F -- E - F-1F = 0, (4.13) n

m = r - l r ~ , ~ s = 1 , . . m . (4 .14 ) r -1 { i f ( x , + y ) } , = l = e~, ,

p = l

Therefore, taking into account (4.7), we have

P~(x~+y)=O.(x~+y)+ -~ . , r-'~{lf(x~+y)},=~ p = l

= Q n ( x ~ + y ) + { - ~ - - ~ , ~ , g ~ } = F ( x , + y ) , s = 1 , . . . , m ,

where ~'~ E R ~ is the vector whose sth coordinate is equal to 1 and the rest of which are zeros. The theorem is proved. []

T h e o r e m 4.5. Let the conditions of Theorem 4.4 hold. Then for all Qn(x) E I'I~(F) formula (4.7) describes the whole set of interpolational c-polynomials in the vector space X at the nodes

{z i+y}~=l C X for all y e Zo.

The proof employs relations (4.13), (4.14) and repeats, for the most part, the proof of the corresponding Theorem 4.3.

Consider the matrices

m

v = ~ l f ( ~ ) , 0 ~ = 1, rk V = r~ - r, r > 0, p=O i , j= 1

Zv = 11~1, ~ ~,ll ~ ,

where c'i, i = 1 , . . . , r , are linearly independent eigenvectors of the matrix V, corresponding to the zero eigenvalue. Reasoning in the same manner as in the case of matrices F and Z, we obtain the corresponding results.

L e m m a 4.5. For any Pn E Ha, the equality

z v ~ . = - d ~ Y"

is valid.

2491

T h e o r e m 4.6. For the polynomial operator interpolation problem (4.1) to be solvable in the vector space X, it is necessary and sufficient that the condition

Z v b :t = -6 E Y~ (4.15)

hold. If this condition holds, then in the case where Qn runs through 1-In the formula

P~(x) = Q n ( x ) + ( ~ - ~ , v - f i { lP (x )}~ ,} p=0

(4.16)

describes the whole set of interpolational polynomials of the nth degree for F in the vector space X satisfying conditions (4.1).

Theorems 4.2-4.5 are also valid in terms of (4.15) and (4.16). Let II be the vector space of operators mapping X into Y; let X and Y be vector spaces; and let

II,~ C II. Further, let F: X --+ Y be a given operator, {xi}iml C X. It is required to find the operator P in the set H satisfying the conditions

P(xi) = F(xi), i = 1,. . . ,m. (4.17)

By analogy with Theorem 4.6, the following result is valid.

For the operator interpolation problem (4.17) to be solvable, it is necessary and sufficient that there exist an operator Q E II for which the equality

= -6 v" (4.18)

holds. If condition (4.18) holds, then in the case where Q runs through l-I the formula

(..~ n m } Pt(x) = Q(x) + - ~ , V - ~ {l~(z)},=,

p=0

describes the whole set of operator interpolants belonging to 1-I.

Let m = n + 1 and let F, Pn: Ra --+ R1. Then the set of polynomials P~(x) in (4.16) consists of a unique interpolational Mgebraic polynomiM of the nth degree of the form

p=O

w h e r e q p = (x~,x2 p, . , x ~ ) , p - - - 0 , . . . , n , 0 ~ 1. Indeed,

V-~- A A ' , A = [1~0~ 1 ...~m--l[], a , = p = 0 , . . . , m - 1,

Then V - ' = (AA*)-' = ( A * ) - ' A - ' ,

A*V -1A = E ~ (ffi, V- ' i f / ) = 5ij, (4.20)

where 5ij is the Kronecker symbol. So, let Q~(x) in (4.16) be an algebraic polynomial of the nth degree of the form

On(x) = ao + alz + ' " + a .x" .

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Then from (4.16) we have

p i n ( X , : Qn(x)-+-(]~c-(ff n, W -1 ~ ~pXP) : Qn(x,+(~ :~, V -1 ~'~rl~xP)--(~n, V -1 ~ ~pXP). (4.21) p=O p=O p=O

Taking into account (4.20), we write the third addend on the right-hand side of (4.21) in the form

V - 1 -., n, x p = a 0 q 0 + a l U , + ' " + a n ~ , , V - ' ffpx v = a o + a l x + ' " + a , x n = Q, (x ) . (4.22) p=O p=O

Taking into consideration (4.21) and (4.22), we get (4.19). Let 1-I1 be the set of linear operators; let F: X -+ Y, and let X and Y be vector spaces. The following theorem is valid.

T h e o r e m 4.7. For a linear operator P satisfying interpolational condition (4.17) to exist, it is necessary and sufficient that the equalities

z =-g ym hold, where Z is the matrix whose rows are coordinates of linearly independent eigenvectors of the matrix

m r : lll,(xj)ll,,,=,

corresponding to the zero eigenvalue, li(x) are linear functionals defined by formula (4.3). Here the whole set of linear interpolational operators can be presented in the form

p I l ( x ) - - QI(x) 21- (~:~- ~1, I~-{li(x)}/~l),

where Q1 E II1 and F - is a generalized matrix inverse to the matrix F.

The proof of the theorem can be readily derived from the proof of Theorem 4.1 for n --- 1 with due regard for the equality PI(O) -- O.

Remark. Note that since this chapter contains a description of the whole set of operator interpolants of a given degree, the results of [126-129] can be interpreted, to a certain extent, as a particular case of the results obtained in this paper.

CHAPTER 3. ANALYSIS OF THE ACCURACY OF APPROXIMATIONS TO POLYNOMIAL OPERATORS IN HILBERT SPACES BY THE INTERPOLATIONAL METHOD

w I n t r o d u c t i o n The study of polynomial systems (see, for example, [125, 106]) has been of great interest in recent

years. First of all, this can be explained by their rather simple mathematical description and, moreover, by the fact that all known analytic identification methods developed for linear systems can also be employed for the case of polylinear and polynomial systems. On the other hand, the study of many nonlinear systems of general form can be reduced to the study of their polynomial approximations. Thus, the polynomial systems theory can be regarded as the connecting link between linear and nonlinear theories. Note that there often occur situations where it is necessary to construct just polynomial models [125]. In this chapter, the investigations from the previous chapter on the construction of Hermite-type interpolational operator polynomials as well as on the analysis of the interpolational accuracy are continued. As is known, the estimates for the accuracy and theorems on convergence are very important aspects in the polynomial operator interpolation theory. Despite the complexity of problems related to the interpolational accuracy in the case of arbi t rary nonlinearity of an operator being interpolated, the results obtained are mainly of theoretical interest and they are of little efficiency in applications. If, after all, we consider the polynomial interpolation of operators of a definite structure, namely, polynomial operators, then we can obtain stronger

2493

(in comparison with the case of arbi trary nonlinearity of an operator being interpolated) results in both theoretical and applied aspects�9

It is shown in this chapter that the accuracy of any approximation to a polynomial operator in the introduced metric can always be increased by the operator interpolation method. The analysis of the interpolation error is carried out, and theorems on the convergence and its rate are obtained (w The operator interpolation is applied to linear operator equations (an example of a two-point boundary-value problem for the second-order ordinary differential equation) (w to problems of polynomial systems identification (w and, also, to finding the trajectory of an object (w

w N o t a t i o n a n d s t a t e m e n t of t h e p r o b l e m Let # be some measure on a separable Hilbert space X such that the corresponding correlation oper-

ator B is a kernel operator and Ker B = O. Consider the Hilbert space Hn(Y) of polynomial operators of the nth degree P,~: X -4 Y:

P,~(x) = Lo + L ,x + L2x ~ + . . . + L , x n, (2.1)

where Lkx k = L k ( x , . . . , x ) = Lk(x) is the kth operator degree [69], Y is a separable Hilbert space with scalar product

n

f X Vk- " " " V, , V k - , " " " V 1 ) k!'~n ~ J Y

x #(dVk) #(dVk_~ ) " " #(dV,)

(2.2)

and norm IIPnl[H,(y) = (Pn,Pn)H,,(y)" Here P(k)(O)VkVk_l . . . V1 is the kth Gs differential at the null-element (9 E X along directions V1, V2, . . . , Vk E X , O! = 1,

�9 Oak Pn aiVi , a i e R1, i -- 1 , . . . , k. P(~k)(O)VkV~:-, "" V1 = 0oq 0a2" . . - i=1 -- oq=o~2 . . . . . o~k=O

Let Pn(x) be a given operator polynomial of the nth degree from H,~(Y) of the form (2.1) and let -Pn(x) E Hn(Y) be its approximation:

Pn(x) = L0 + f-nix .2f_ L2 Z2 . .~ . . . + y_jn xn. (2.3)

With the polynomial Pn(x), and its approximation fin(x), we associate the following interpolational polynomial operator of the nth degree PI,m(X ) e Ha(Y):

pI,m(X) = Lo + L~x + L~x 2 + . . . + L~x '~ (2.4)

with interpolational conditions

Pn ~(0 (O)Bxk, Bxk, 1 "'" Bxkl = P(O(O)Bxk, Bxk,_~ ..�9 Bxk, (2.5) 1112

�9 . m for all 1 < kl ~ k2 <: " '" ~ ki ~ m, i -~- 0 , . . , n, whe re {x ,} i= 1 C X is a linearly independent system of elements. The structure of L~x k, k = 1 , . . . , n, will be determined hereafter. Let us introduce the errors

a ( x ) = ? , , ( x ) - p n ( x ) ,

a m(X) = P ,m(X) - P , ( x ) .

(2.6) (2.7)

The main purpose of this chapter is to analyze the polynomial error OIm(X) in the metric of the space Hn(Y) and to compare it with the error O(x) in this metric. In so doing, it will be shown that for any approximation /3,~(x) to the operator Pn(x), it is always possible to construct an interpolant P~,m(x)

for Pn(x), which depends on /3n(x), with higher approximation accuracy (even if only one interpolational node is used).

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w Auxiliary results Let us formulate some lemmas, which will be needed for proving main theorems of the next section. Consider the vector

( 3 . 1 ) = {(X, Xil)X(X, Xi2)X ' ' ' (X , Xik)X for a l l l < i l < i 2 < . . . < i k < m } Mk = C k

. . . . . ~ m + k - - 1 "

E x a m p l e s . For k = 1, we have

l l (Z) : {(X, Xl)X(Z, Z2 )X , . - - , (Z , Xm)X}.

For k = 2, we have

h(z) = {(z,z1)x(z,z1)x,(z,z•)x(z,z2)x,...,(•,z1)x(z,zm)x,(z,z2)x(•,z2)x,(•,z2)x(z,•3)x, (x, z2 )x ( z , x m ) x , . . . , (x, xm-1 )x(z , Xm--1 )X, (Z, xm-a )X(X, xm)X, (X, z,~)X(Z, Xm)X }. O I Q ~

L e m m a 3.1. Let the set of indices 1 <_ kl <_ k2 <_ " . <_ ki <_ m correspond to the kth coordinate of the vector ~(z) and let the set of indices 1 <_ s~ <_ s2 <_ ... <_ s~ <_ m correspond to the sth coordinate of this vector. Then the relation

1 (i) 1 l(i)/A~T , (tki,lsi)H,(R~) = ~.lki (| Bx~,_~ . . .Bx~, = -ft. ~i (~)~xk , Bxk,_~ ...Bz~:~, k , s = 1 , . . . ,Mi ,

is valid.

Proof. We have

1 0 ~ (lki,lsi)H,(RD - (i]) ? f x " " s OalOot2...Ooti

i i i

oi i i i

X oa~o;'"aai {Q~-_~a~VJ'z'~)x (i~=laJV~'x'~)x "'" Q~=~aJVi'xS')x}t~ . . . . . ~ , = 0

• I-t(dVi ) Ig(dVi_l ) . . . ~(dV1 )

- "

Jl # j2r "7~ ji

1

jl C j2#"'5~ ji jk=l,2,. . . , i

jk=l,2,.. . ,i

•

jl r162 jh=l,2,...,i

X/z(dI/i) pt(d]/~-i ) . . . #(dV, )

1

jl ~ j 2 ~ ' " ~ ji jh=l,2 ..... i

The result of Lemma 2.1 from Chapter 1 was used in these transformations. Lemma 3.1 is proved. []

2495

L e m m a 3.2. I f xi, i = 1 , . . . ,m , are linearly independent, then lik(x), i = 1 , . . . , M k , are also linearly independent for all k = 1 , . . . , n .

Proof. Let lik(X), i = 1 , . . . , Mk, be linearly dependent . Then there exist constants cl , c 2 , . . . , CMk, not all zero, such tha t the equal i ty

cl l lk(x) + c212k(x) + "." + CMkIMkk(X) = 0 (3.2)

is valid almost everywhere in X. Consider the span/2 of the d e m e n t s x l , x ~ , . . . , x , , . It is known [69] that �9 r / q any element x E X can be represented in the form x = y + z, where y E/2, z 5_/2. Let {Y,}i=l be a system

of elements f rom/2 bior thogonal to {x,}i=a. Such a system always exists. Indeed, let

Yi = al jXl q- a2jx2 q- "'" q- otmjXm, j = 1 , . . . , m,

and let the condit ion ( x i , y j ) x = 6 0 hold, where 6ij is the Kronecker symbol. T h e n for finding coefficients akj , k = 1 , . . . , m, we get a linear system of elements with Gram mat r ix }f(xi, which is

�9 m nonsingular in view of the linc..r independence of tt.~ system {x,}i=l. Since yj, j = 1 , . . . , ra, are linearly independent elements, any element y E/2 can be represented in the form

Y = fla Yl + f12Y2 q - " " q-

Set x = y + z in (3.2), where y is defined by relation (3.3) and we get

�9 . . C k C1/31 k -~ C2fll k-lfl2 -~- -[- Mkflrn ~- 0 for

flmYm. (3.3)

z _L s Then, taking into considerat ion (3.1),

al l fl~, f12, . . . , #m E/~1. (3.4)

On the lef t -hand side of (3.4), we have the k th form of m var iables/31, /32, . . . , tim, equal to zero for any values of these variables. The la t te r is possible only with cl = c2 . . . . . cuk = 0, which contradic ts the linear dependence of lik(x), i = 1 , . . . , Mk. L e m m a 3.2 is proved. []

�9 O O L e m m a 3.3. Let the system of elements {X,}i= 1 be complete in X . Then the sequence of functionals lik(x), i = 1 , 2 , . . . , is complete in the space Hk(Ra) of k-degree functionals for all k = 1 , . . . , n .

Proof. Let Ik(x) = Lk(x , x , . . . , x) be an arb i t ra ry kth functional degree from H k ( / ~ ) and let

= 0 , i = 1,2, . . . .

Let us show tha t Ik(x) = 0 for all x E X. Taking into account L e m m a 2.1 f rom Chap te r 1, we have

l/x fx{ } ( l ik(x) ' lk(x))Hk(R,) = k-~, "'" E ( V J l ' x h ) x ( l ' ~ 2 ' x i 2 ) x ' " ( V J ~ , ' x i ~ ' ) x L k ( V l ' V 2 " " ' V k ) jl C j2 r Jk

x #(dVk)#(dVk_, ) . . . , (dV, )

= L k ( B x i , , B z i 2 , . . . , B x i k ) = O, k = 1 , . . . , n . (3.5)

O O . O O If {xi}i=l is a complete system of elements in X, then {Bx,}i=l is also complete in X. Indeed, since B is a self-adjoint opera tor , for all x E X we have

(Bx i , x ) x = B )x = O.

From the completeness of {Xi}i~176 it follows that B x = 0, but K e r B = O, therefore x = 0. Thus, the system {Bx , } i= l is complete in X. The lat ter means that for any e > 0 and x E X there exists an element x~ E X, for which

X~

Ilx -- xe}I < E, (3,6)

N

= Z ciBxi , N = N(e) . (3.7 / i = 1

2496

Further,

L k ( x , x , . . . , x ) = L , ( x - x~ -t- x ~ , x - x~ + x ~ , , . . , x - x~ + x~)

: Lk (x -- x~ ,x -- x ~ , . . . , x -- x~) + C ~ i k ( x - x~ ,x - x ~ , . . . , x - x~,x~) (3.8)

+ C~Lk(x - x~ ,x - x ~ , . . . , x - x~,x~,x~) + . . .

f ' k - - I i t x + ~ k kt - z ~ , x ~ , z ~ , . . . , x ~ ) + L k ( x ~ , z ~ , . . . , x ~ ) .

Taking into account (3.5) and (3.7), we have L k ( z ~ , x ~ , . . . ,x~) = 0. Since Lk(x) are continuous (see w Chapter 1), they are bounded; hence from (3.8) we have

iL, (x)l _< ilLkii{llx-x iiX, + llxllx, li -' }, k = 1 , . . . , , , . (3.9)

On the basis of (3.9), the r ight -hand side of (3.6) can be made arbitrarily small for any x E X. This is I ' evident, since IIx~l(x = Itx~ - x + xl tx <_ IIxllx + e. But for all x E X and for any arbitrarily small poe ' : i re

e, the inequality [Lk(x)l < e holds, independent ly of x, only in the case where Lk(x) = 0 for all x E X. The l emma is proved. []

�9 oo complete system of elements in X and let {yv}~=~ be an orthonormal basis L e m m a 3.4. Let {x,}i=l be a in Y . Then the sys tem of elements

k,p= 1,2,..., (3.to)

is complete in the Hilbert space H n ( Y ) of the nth degree operator polynomials.

Proo/. Let P=(x) = Lo + LlX + ' " + L . x ~ be an operator polynomial from H n ( Y ) and let

( I i~(x)yp ,Pn(X))H,(y) = 0 , s = O , . . . , n ; i = 1,2, . . . . (3.11)

Let us show tha t in this case we have Pn(x) = 0 C Y . "raking into account the scalar p roduct in the space H ~ ( Y ) defined by formula (2.2) as well as the form of lis(X) irl (3.1), from system (3.11) we get the following system of equalities:

(yp ,Lo)v = 0, p = 1 , 2 , . . . , (3.12)

(li,(x)y,,L,(x)),,(v) = (yp,L,(Bxi))g = O, i = 1 , 2 , . . . , p = 1 , 2 , . . . , ( 3 . 1 3 )

( l i2 (x)yp ,L2(x) ) ,2 (y) = ( y , , L ~ ( B x i , , B x i ~ ) ) y = O, i , , i 2 , p = 1 , 2 , . . . , (3.14)

( l in(X)yp,Ln(X))H, , (y) = ( y p , L n ( U x i ~ , . . . , B x i , , ) ) y = 0 , i i , . . . , i n , p = 1,2, . . . . (3.15)

The result of L e m m a 2.1 from Chapter 1 was used in t ransformations of formulas (3.13)-(3.15). Since yp, p = 1, 2 , . . . , is an o r thonormal basis in Y, from the last equalities it follows that

L k ( B x i ~ , B x i ~ , . . . , B x i k ) = 0 E Y, k = O , . . . , n , i l , i 2 , . . . , i k = 1 , 2 , . . . .

Further, reasoning in the same manner as in the proof of Lemma 3.3 with the replacement of the function lk (x) by the opera tor Lk(x ) and taking into account the completeness of the system B x i , i = 1, 2 , . . . , in X, we get

Lk(z ) = 0 ~ Y, k = 0 , . . . , n, x E X .

Thus, we arrive at the conclusion that equality (3.11) with s = 0 , . . . , n and i = 1 , 2 , . . . can be fulfilled only in the case where P~(x) = 0 E Y . Lemma 3.4 is proved. []

2497

L e m m a 3.5. The equality

is valid.

tl 2 IILkllH.r) (3.16)

k=O

The proof follows directly from the definition of the norm in Hn(Y) and representation (2.1).

w M a i n t h e o r e m s of t h e a n a l y s i s of a c c u r a c y of t he p o l y n o m i a l o p e r a t o r i n t e r p o l a t i o n Before presenting main results of this section, let us define the operator polynomial P~n,m(X) in (2.4)

and show that interpolational conditions (2.5) are fulfilled for it. By analogy with [46], we set

L~x k : LIk(x) = L k ( x ) - ~<&,F~-l~k(x)),. k = 1 , . . . , n ;

Ok = {O(k)(O)BxikBxik_l " " B x i l , for all l _<il _<i2 < - - - _ < i k < ira},

(4.1)

and

Ok(x) = I . k ( x ) - Lk(x);

k rk is the Gram matrix with elements (lik,ljk)Hk(R1), i , j = 1 , . . . ,Mk, Mk = Cm. t_k_ l , the vector /'k = {llk(X),12k(x), . . . ,lM~k(X)} is defined by relations (3.1);

Mk ('} = 2 0 l i ~ i '

i=1

E x a m p l e s of vectors Ok. For k = 1, we have

ai E Y, ~ E Rx.

O1 : {O~(O)Bxl,O[(O)Bx2,. . . ,O~(O)Bxm}.

For k = 2, we have

02 {O•'(O)BxlBx, "" = ,O; (O)Bx2Bz l , . . . " " , O; (O)BxmBx, , O;'(O)Bxa O; (O)Bx2 Bx2, Bx2, . . . ,

O;'(O)BxmBx2,.. ,O;(O)Bxm-1 ,t �9 " B x m - , , O ; (O)BxmBxm_, ,O; ' (O)BxmBxm . J

T h e o r e m 4.1. The nth-degree operator polynomial P�88 in (2.4), the kth degrees of which are defined by formula (4.1), is a Hermite interpolational polynomial for Pn(x) with interpolational conditions (2.5).

Proof. On the basis of (2.4) and (4.1), we have

I(0 P~,m(O)Bxk, Bxk,_1 "." Bzk, = LIi<i)(O)Bxk, Bxk,_, . . . Bxk, = Lli)(O)Bxk, Bxk,_,

1(5i , i~?1{ J~xk,_, /~Xkl}8_~l/ i! Z )(O)Bxk, . . . M,

Taking into consideration the result of Lemma 3.1, we get

. l~ii)(O)Bxk'UXk' ' "'" i s = l )

"'" Bxkl (4.2)

= (5i, Pi-l{(lsi, lki)Hi(Rl)}sMi=l)

= , P'~lFigk = ,gk =O}i)(O)Bxk, B x k , _ , ' " B x k ,

= Lli)(8)Bxk, Bxk,_, "-" Bxa, -- L~i)(8)Bxk, Bxk,_~ . . . Bxk, . (4.3)

2498

Here gk E RM~ is a vector, the kth coordinate of which is equal to 1 and the rest are zeros. Also, note

that the set of indices k l , k ~ , . . . , ki corresponds to the kth coordinate of vectors ~/ and ~ . Taking into account (4.2) and (4.3), we get conditions (2.5). The theorem is proved. []

Now let us turn to the presentation of the main results of this section and of the chapter as a whole. The analysis of accuracy of the polynomial operator interpolation in the metric of the space H ~ ( Y ) is connected with the analysis of the error

Olin(X) ~-- p I , m ( X ) -- P n ( x )

and, respectively, with the relation between the errors OIm(X) and

n

o(~) = ~ . ( ~ ) - p . ( ~ ) = ~ O~(x) k=l

in this metric.

�9 m T h e o r e m 4.2. Let a system of elements { X , } i = 1 be a linearly independent system in X and let Bx i , i = 1 , . . . , rn, be a sequence such that at least one of n conditions

oi~)(O)Bx,~ Bx,k_, . .. B~,, = k! O , ( B ~ , , , B x , ~ _ , , . . . , B ~ , , ) # O, 1 <_ il <_ i2 <_ . . . <_ ik <_ rn, k = 1 , . . . , n ,

(4.4)

holds. Then the equality

is valid.

II0~IIH.(~) < II011H.(Y) (4.5)

Proof. Put o~,~ = L I ( ~ ) - L~(~), k = 1 , . . . , n . (4.6)

Then, in view of Lemma 3.5, in order to prove (4.5), it is sufficient to show that

IIoL~ll~.~) ~ IIOkIIH~(Y), k = x , . . . , ~ , (4.7)

and if condition (4.4) is fulfilled for a fixed k, then (4.7) becomes a strict inequality for this k. Taking into consideration (4.1), we have

1 Mk

Ol'k(X) = Ok(X) -- -~. E "[}# k)Oikljk(x)' i,j=l

(4.s)

(Mk) where ~/ij are elements of the inverse matrix F~ -1, Oik is the ith coordinate of the vector Ok, and ljk(x)

is the j t h coordinate of the vector lk. Further we find

~ _ 2 M. (Uk)(O~(X)'O'kbk ' ' - - - - , i , j=l

Mk 1 M~ (Mk) (Mk)

i,j=l s,t=l

(4.9)

2499

Let the set of indices 1 _< il _< i2 <_ . . . _< ik < m correspond to the ith coordinate of Oik in the representation of the vector Ok, and let the set of indices 1 < j l _< j2 _< "'" <_ jk _< m correspond to the j t h coordinate of ljk(x) in the representation of the vector/'k. Then

(&(x), O~G-k (x)) re(r) = (Ok(x)'k!Ok(Bxi~'Bxi' ' ' ' ' 'Bxik)ljk(x)) H~(Y)

• ~ (",, ~s, ) ~ (v~, x , ) ~ . . . (v~, x,. )~,(dVk),(dYe_,)....(d.~)

: k! ( O k ( B x i , B x i 2 , . . . , B x i u ) Ok(Bz j , B x j = , . . . B x j k ) )

(4.10)

The result of Lemma 2.1 from Chapter 1 was used in transformations (4.10), and in the sum Z summation is over all permutations of variables V1, V2,. . . , Vk. Taking into account (4.10), we rewrite the second addend in (4.9) in the form

i , j=l

Mk

i , j=l

(4.11)

It is not difficult to notice that

M k

7ij k J x , Irk j = l

"= ~it ,

where (~it is the Kronecker symbol. Taking this fact into account, we write the third addend on the right- hand side of equality (4.9) in the form

Mk Mk 1 (Mk) (Mk)(Oikljk(X),O~kltk(X))

i , j=l s,t=l

M~ (M") (O'k 'Otk)Y - 1 M. (M' ) (Oik 'OJk)Y 1 M. Mk (Mk) (Oik 'Osk)Y ~it _ 1 E %t (k!)2 E 7ij -- (k!)~ Ei=I s,t--=lE "/st \ (k!)2 s , t=l i , j=l

Mk

- ~rij ~.~'k(Bzi,,Bxi=,... ,Bzik),Ok(Bzk,Bzj=,... ,Bxjk) y. i,j.=l

(4.12) Ol 2 Now, from formulas (4.9), (4.11), and (4.12), we finally get the following expression for II m,kllH.(y):

[ l o g ~ =

Mk

(Mk) (Ok (Bxit llOktl~,.y)- ~ ~,,j i,j=l

,Bxi=,... ,Bxi~),Ok(Bxj,,Bxj=,... ,Bxj~))~; (4.13)

Let {Yp}~=l be an orthonormal basis in Y and let

oo

Ok(Bzi ' 'Bxi2' ' ' ' 'Bxik) = E ~ipyp, p=l

~ip E t/1.

2500

Then Mh

"Tij , , " " " , , , Y i,j----I

o~ Mk (4.14) ( M k ) r f : . _ _ = ..

= E E "yi j ,~,p,3p > O, k l, . ,n , p----1 i , j = l

since Fk I is a positive-definite matrix. If, after all, condition (4.4) is fulfilled for some fixed k, then for this k, (4.14) turns into a strict inequality. Taking into account the last remark as welt as formulas (4.13) and (4.14), we get (4.7), which completes the proof of the theorem. []

This theorem has a simple but extremely important interpretation, namely: for any approximation Pn(x) to the polynomial operator P,~(x), it is always possible to improve its accuracy in the metric of the space H~(Y) with the help of operator interpolation.

Denote by X a completion of X_ in the norm I1~11- = (Bx, x)~ ~. T h e o r e m 4.3. Let a system of elements {Xi}i~176 be linearly independent. Then the inequality

[IO~m+allH~ <_ IIo~IIHo(~) (4.15) �9 OO is valid for all rn = 1,2, . . . . If {x,}i=l C X is an orthonormal system in X _ and at least one of the

conditions (4.4) is fulfilled, then (4.15) becomes a strict inequality.

Proof is based on the result of Lemma 3.5 and, respectively, on inequality (4.15), where the error O~(x) in the norm of H n ( Y ) is replaced by the error O~,k(X ) in the norm of Hk(Y) .

Let {yp}~~ be an orthonormal basis in Y. Taking into account Lemma 2.1 from Chapter 1, we transform the relation (4.13) in the following manner:

I~m,~llH.Y) Io~(Y,,Y2, .,Yk) ~(dY~)~(dY~_,) ~(dY~) M~ o~

i , j = l p----1

oo

. . . . / zr , . . y p ) y # ( d V k ) # ( d V k _ l ) . . . # ( d V 1 ) p = l

co Mk

p = l i , j = l

Y

Under the conditions of Theorem 3.7.9 from [78], the series

II0~(v,, y ~ , , v~)ll~ = ~ (0k(v~,v2,, yk),yp)~ p = l

can be termwise integrated, since

N

(Ok(V,,W2, .,Wk),yp)~ < Ilok(W,,W~,. ,w~)ll~ p = l

for all N

]g ~ 1 , . . . ~ n .

(4.16)

(4.17)

2501

and the r ight-hand side of this inequality is integrable. The equality

(Ok(Bxil , Bxi2,.. . , Bxik ), yp) V i = 1 , 2 , . . . , M k ,

follows from Lemma 2.1 in Chapter 1. Our further reasoning is as follows. The independence of elements {Xi}i~176 1 provided for by Lemma 3.2

implies the independence of/~k(x), i = 1 , . . . , Mk. Therefore the expression in curly brackets in (4.16) is the squared distance between the point (Ok(x), Yp)v and the span of elements Ilk(x), 12k(x),..., IMkk(Z) in the metric of the space Hk(R1), and hence this quantity does not increase in m. Thus the inequality

_ k = 1 , . . , , ~ , (4.1S)

and, respectively, inequality (4.15) are proved. Now let {x,}i=~ be an orthonormal system in X_ and let at least one of the conditions (4.4) hold.

Then the strict inequality in (4.18) and (4.15) immediately follows from the relation

, 2 -IloklIH. ) II0m,kll. ( ) -

m

E COk(BXil'Bxi2'""Bxik)ll ; ' (4.19) il , i2,. . . , ih=l

which, for its part, is obtained from (4.13) if {xi}i~176 is orthonormal in X_. The theorem is proved. []

T h e o r e m 4.4. Let {xi}i~=l be a complete system in X. Then the interpotational process PI,m(X ) converges in the metric of the space Hn(Y), i.e.,

IIo m I I . . y ) = o. (4.20)

Proof is based on relation (4.16) and Lemma 3.5. By the theorem and Lemma 3.3, the sequence lik(x), i = 1 , 2 , . . . , is complete in the space Hk(R1) of k-degree functionals and hence it is closed. Therefore, on the basis of generalization of the closedness equation [2], the limit of the expression in curly brackets in (4.16) as m ~ co is equal to zero. It remains to substantiate the passage to the limit under the sum sign of the series on the right-hand side of (4.16). But the series in (4.16) is a series uniformly convergent in m since it is majorized by a convergent numerical series (independently of m). Indeed,

(ok(x),yp)y,,Uh(R,) = , .. yp)yp(dVk)#(dVk-,) . . .#(dV1)

oo

. . . . , . . . , yv)y#(dVk)#(dVk-1). . .#(dVl) p = l

= l l o k l l ~ ( g ) . (4.21)

Here the legitimacy of termwise integration of series (4.17) is used. Therefore, series (4.16) converges uniformly in m and the passage to the limit as m -+ co involves the relation

l i m II0' ,kll.~(y) = 0 , , k = 1 , . . , n . rrt--~. r

(4.22)

By Lemma 3.5 from (4.22), we immediately obtain (4.20). The theorem is proved. []

2502

T h e o r e m 4.5. Let {x,}i= 1 be an orthonormal basis in X composed of the eigenelements of operator B. Then the estimate

is valid, where a = a(B) > 0 and C is a positive constant, independent of m. Proof. Denote by hi > 0, i = 1 , 2 , . . . , the eigenvalues of the operator B. Then under conditions of the theorem from (4.13), we find

]lOm,kllg~(y) . . . . Ilok(Vl,V~,...,v~)lly#(dV~)#(dV~_,)...,(dV, ) m (4.24)

E il,i2,...,ik=l

Let us represent the variable Vp E X, p = 1,..., k, as a Fourier series:

Vp : E(Vp,xi)xxi, p = 1 , . . . , k . (4.25) i = 1

Taking into account (4.24) and using Lemma 2.1 from Chapter 1, we write the first addend on the right-hand side of (2.24) in the form

�9 . . v )llY.(dV ) #(dVk-i ) " " ~(dVl )

: fX'''fXl{ ~ (Vl,xil)x(V2,xi~)x'"(Vk,xi.)xOk(Xil,xi2,'",Xi~,) t ~ il,ix ..... ik=l (4.26)

x ~,(dVk) # (dVk_ , ) . . . ~,(dV,) o O

il ,i2,...,ik=l

Under the conditions of Theorem 3.7.9 [78], the above series may be integrated by terms since

11 I1 fX"" fX (Vl,Xil)x(V2,xi2)x"'(Vk,xik)xCgk(Xi~,xi2,...,zik) #(dgk)#(dVk-1)'"#(dV1) Q ,i2,...,ik:l y

N N N < [[OkII2/X'''/X E (Yl'Xi')x E (Y2'xi2)x"" E (Yk'xik)2 _ x~(dVk)#(dVk_~) . . . ~(dV~)

i 1 -=1 i 2 = 1 ik=l

< ]]0kl]2 fx ]]VIII2#(dV1)fx 11V2]I2#(dV2)"" Ix ]]Vkn2#(dVk)< cx~ for all N.

Taking into account (4.26), we represent the equality (4.24) in the form

2 It0LkllH (y/ =

il ,i2,...,ik=l r F t

il,i2,..,ik=l

il=m+l i2,i~,...,ik=l

hi, khk-'ll&ll 2 ~ k = l , . . . ,n, i = m T 1

(4.27)

2503

where ,k = max ,~i. Since B is a kernel operator, the series E,~i converges and hence

C )~i ~< i1+--'-- ~ , (r > 0 , C = c o n s t >0 .

Then, taking into account (4.27), we have

O O

I 2

i=m+l

c k = Ck, k-lll&lt2, k ---- 1 , . . . , n .

= Ck(om")- ' , (4.28)

And, finally, taking into consideration (4.28) and Lemma 3.5, we get inequality (4.23), where C = ~2=~ Ck. The theorem is proved. []

Consider the span s in H,(Y) generated by the system of elements

{lk,(z)yp }, k = 1 , . . . ,Ms , s = O , . . . , n , p = 1 ,2 , . . . , M o = l .

Any element from s can be represented in the form

n Ms N

t~ks ,ks(X)y,, �9 R1, Ot ks s=0 k = l p = l

(4.29)

where N is an arbitrary natural number, dependent, in general, on the element under consideration. Denote by s the closure of s

T h e o r e m 4.6. The orthogonal projection Pn(x) �9 gn(x) onto s coincides with the Hermite interpolational polynomial PI,m(x ) for Lk(x) = O �9 Y, k = 1, . . . ,n.

Proof. First we show that P~,m(x) �9 s for Lk(x) = O, k = 1, . . . ,n. Setting

M,

ks =- E l k i Ls ( B x i l , Bx i2 , . . . , B x i , ), yp Y i=1

in (4.29) and substituting their values into (4.29), we get

n M , N

E ~ks lks(X)yp s=0 k = l p = l

N n M~ N ~ - ~ (p) (P)

p = l s---=l k = l p----1

N n M, N M~

p = l s = l k--=l p--=l i = l

N n M, N

lks(z) ,exi2,..., p = l s = l i , k = l p = l

n Ms

-"+ L~ q- E Z " Y ~ M ' ) l k s ( X ) i s ( B x i " B x i 2 " ' " , B x i . ) as N -+ exp. (4.30) s = l i,k=l

But the limit in (4.30) coincides with PI,,m(x ) for Lk(x) = 8, k = 1, . . . ,n, meaning that P~,,,,(x) �9 -s if this condition is fulfilled.

2504

Now we show that Pi,,m(x ) for Lk(x) = 0, k -- 1 , . . . , n, is an orthogonal projection onto the space ~(rn). We have

n M. n M, N

s = l i ,k=l s=0 j = l p = l H . ( Y )

= L~(x ) - E " f }M~)4~(x)Lr(Bx '~ 'Bx i~ ' ""Bx '~) 'EEa~)4~(x)YP r = l i ,k=l s = l p = ] H~ Y)

= E E , IP~ ) L~(Bx~,,Bx~=,. . . ,Bx,~),y, y r : l s = l p-~-I

M~ Mr N

- - ~ ~ "/ik s r ~ "'" i ,k=l s=l p=l

= a(2. ) L , . (Bx,~,Bx,=, . . . ,Bx, , ) ,yp y r = l s=l p=l

M, M~ Mr N

-- 7 ik I, t k r , l s r ) H r ( R 1 ) E O l ! p) L r ( g x i l , B x i 2 , . . . , g x i , ) , Y p y i=1 s = l k = l p = l

( M~ sr �9 �9 �9 N

Mr Mr N

i = l s = l p = l

= z_~S-" ~(P)~ L~ (Bx~1, Bx,= , . . . , B x s ~ ) , yp Y r = l -- p = l

M. N zz=( )):o _ ~(p) L ~ ( B x , I , B x , 2 , . . . , B x ~ ) , y p y s = l p = l

Here the identity Mr

k = l

was used, where ~is iS the Kronecker symbol. The theorem is proved. []

As was noted above, the interpolants constructed in Chapters 1 and 2 have the extremal properties since they are solutions of certain extremum problems. From Theorem 4.6, it follows that for Lk = O, k = 1 , . . . , n , the interpolationat polynomial operator P~,m also represents a solution of the extremum problem

Pn - p I , m = inf liP. -- QnIIH.(Y) for all P. E Hn(Y). H , ( Y ) Q , E~(m)

w A c o n n e c t i o n b e t w e e n t h e n o r m s II'll+, II'IIHI(Y), a n d I[']l of a l i nea r o p e r a t o r In this chapter, the polynomial operators Pn were regarded as elements of the Hilbert space H~(Y).

Clearly, linear operators may be regarded as elements of the Hilbert space H1 (Y). The study of connection between the traditional norm of a linear operator and its norm as an element of the space H1 (Y) is of certain theoretical and practical interest (see the next section).

Let us introduce a new scalar product in X:

(x,y)_ = (Bx, y)x = (B1/2x, BU2 y)x.

2505

Denote by X the completion of X_ with respect to this scalar product (it can be regarded as an expansion of X). The space X is an everywhere dense subset in X_. Denote by X+ the Hilbert space obtained from the domain of definition of the operator 13 -1/2 (it is dense in X) by introducing the scalar product

(x,y)+ = ( B - i x , y )x = (B-1/2x, B - ' 1 2 y ) x .

The operator B can be continuously extended to X_. In this case the relations

B i / 2 x _ = X, B1/2X = X+, B X _ = X+

hold, while the positive and negative spaces X+ and X_ constructed from X and the operator Bll2: X X+ form the chain X+ C X C X_.

Let the linear operator L1 be continuous in X, L I : X -+ Y. Then in the spaces X+ and X its norms are defined by the relations

and

IlL, N+ = sup l lL,(x) l l r , x E X+, (5.1) NxN+=I

IIL, tl = sup IILl(x)l ly, x c x , (5.2) NxH=l

respectively. There exists a fairly simple connection between traditional norms (5.1) and (5.2) of a linear operator and its norm as an element of the Hilbert space Hi (Y) .

T h e o r e m 5.1. The inequalities

IILIlI+ < t lLl l lm(Y), I IL l l l . , ( r ) <_ IIL, II (Sp B) ~12

(5.3) (5.4)

are valid, where oO Sp B = f . I1~112 ~(&) = ~ ~ < (30

d.A i=1

and )~i are eigenvalues of self-adjoint positive kernel operator B. In this case, if LI: X --~ R1, i.e., if L1 is a functional, the inequality (5.3) turns into an equality.

Proof. Let {yp}~=l be an orthonormal basis in 12. Then

oo H/1H~- / I (Y) ---~ i x Hil(x)ll~#(dx) ~- ix ~_ (LI(x)'yP)y#(dz)

oo = F, fx(a,,x)}"(dx), a, e X , p = 1,2, . . . .

p-~l

(5.5)

Under the conditions of Theorem 3.7.9 from [78], the series may be integrated by terms since

N

p=i

for all N, (5.6)

while the integral on the right-hand side of inequality (5.6) exists as Ll E H1 (Y). Next, we have

cx3

IIL, II~,<y) = ~ ix(ap,x)x(ap,X)x ~(dx) = p=l p---=l p = l

(5.7)

2506

On the other hand,

IIL, II~_= sup IILI(x)H~= sup E ( L , ( x ) , y , ) ~ tlxll+=l II*ll+ =1 p=l

= sup E (b"x)2+ -< }--~ l[/'p[t~-, b, E X + , p = 1,2, . . . . Hxll+=l p = l p = l

(5.8)

But since

we have

( b y , x ) + = (av,X)x for a l l x E X + , p = 1 ,2 , . . . ,

bp, x ) + -

Put B-1/2x = y E X. Then

(ap,x) x = ( t ~ - I / 2 b p ' B - 1 / 2 x) X - - ( a P ' x ' ) x • O.

(bp,*)+ - ( - , , x ) x = (B-a/2bp ' y ) x - - (aP 'B1/2y)X

= (B-1/Zbp - B l /2%,y )x = 0 for all y E X. (5.9)

From (5.9) we immediately find

and hence bp = Bay, p = 1,2, . . . . From inequality (5.8) we get

B-a/2bp - B1/2 ap = 0 E X,

][LllI2+ <_ EHbpll2+ = E ( b p , bp)+ = E ( B - a b v , bp)x = E ( B a p , a p ) x = Z [[avll=.. p = a p = l p = l p = l p--1

(5.1o)

Comparing (5.10) and (5.7), we make sure" of the validity of inequality (5.3). Note that in view oo 0o of (5.10) the series 2 , = 1 Ilb, l[~- converges due to the convergence of the series Y'~-p=a Ila, ll%. The latter

follows from (5.7), since L1 E Hi(x) . If L I : X - R 1 , i.e., it is a functional, then

H/1H~/'(Y)-- f x ] lLl(x)H~z"(dx)= ~ ( a , x ) 2 #(dx) = (Ba, a)x = Ila]l 2,

liLlll~--- sup lLl(x)12= sup (b,x)2+=llb]12+, b ~ X + . tlxll+--1 Ilxlt+=l

a E X , (5.11)

(5.12)

But since b = Ba, the comparison of (5.11) and (5.12) allows us to make sure that in this case inequality (5.3) turns into an equality.

Next,

HLaJig,(y) : [JLa(x)[J~ ~(dx) ~ llLaJl 2 I[xlJ 2 ~(dx) : HLaII 2 ~ A, = tlLa][ z SpB, i=1

which proves inequality (5.4). Theorem 5.1 is proved. []

Remark. As a rule, in practical problems, the estimation of the operator 's value at a point (for example, the estimation of IlL1 (xo)llY) is of interest. If x0 C X+, then on the basis of the previous theorem, we have

IILl(xo)llY ~ IILIH+ IIxoll+ ~ HLIHHt(Y)Ilxoll+.

2507

Moreover, under the conditions of Theorems 3.2-3.4, the interpolation error OIm(x) = L~ (z) - L~ (x) of the linear interpolational operator

L~(x) L(x)

satisfies the relations

IIO, llmcY) > Ilolll.,< 11'11 o, lim Om H,(Y) =

r r t - - ~ o o

01 >11 O,(x) = L l ( x ) - Ll(x).

Note that some questions related to the approximation of values of a linear operator in Hilbert spaces were also studied earlier (see, for example, [67]).

w A n a p p r o x i m a t e so lu t ion of l inear o p e r a t o r e q u a t i o n s by t h e i n t e r p o l a t i o n m e t h o d The results of this chapter can be used in solving linear operator equations in Hilbert spaces. For

example, consider the problem

A u = f , f E X , u E Y ,

�9 m where X and Y are separable Hilbert spaces and A is a linear invertible operator. Let {u,}i=l be linearly independent elements from Y, on which the values f i = Aui, i = 1 , . . . , m, are known ( f i are also linearly independent in X), and let the operator B be such that the equations

B~= f,, i=1,. . . , ,~,

can be easily solved. Then, if some approximation fi,-~ to the inverse operator is known, then the procedure (2.4) considered in this chapter for n = 1, L0 = 0 E Y makes it possible to construct an approximation to the inverse operator in the sense of the metric of the space HI (Y).

E x a m p l e . Consider the following two-point boundary-value problem for a second-order ordinary differential equation:

Au = u" - xu = - f ( x ) , x e (0,1), u(0) = u(1) = 0. (6.1)

Put

M(,)=fo' where

{x(1 - ~1, a(x,~:) , , ( l -x ) ,

Let X = L2(O, 1). Then the positive and negative spaces

a(x, ~)/(~) d~,

x<~, ~<z.

o

X+ = W21(0,1) and X_ = W2-1(0,1)

constructed from X and the operator B form the chain

o

W~(0, 1) C L2(0, 1) C W2-1(0, 1).

3 i t t Letui(x) = I x ( l - x ) ] x , i = 1 , . . . , m . T h e n f i = - u i + x u i , i = l , . . . , m . B f i = - f i a r e s o l u t i o n s o f t h e equations ]i = fi", / = 1 , . . . , rn; the quantities FI are elements of the Gram matrix ( B / i , ]j ) = - ( f~ ' , f j ) = ( f [ , f } ) , i , j = 1 , . . . , m ; and ( - , . ) is the scalar product in L2(0,1).

2508

For approximate solution of Eq. (6.1), a recurrent procedure of the form

L ~ , , n ( - f ) = L I (01 ~ l ( - f ) ) l , m - l ( - f ) - ,Y71 , m = 1 , 2 , . . . ,

L I : O, 1,0

where

(6.2)

OIm(x) = L I ,,m-l(X)-Ll(x),

51 = (OI(B/1),OI(B/2), . . . ,OI(B/m)) ~-- (O,O,...,O, Ll , rn_l( - fm)- t t rn) ,

r , ( - s ) = =

is chosen. Moreover, for comparison with the solution of Eq. (6.2) obtained by recurrent procedure (6.2) for L I = 0, the solution obtained by the same procedure but with L I = B f was studied. It is clear that 1,0 1,0 B f is also a solution of the problem

I! u = - f ( x ) , x C (0, 1), u(0) = u(1) = 0.

The results of computa t ions are as follows:

U* ~ s i n 71"x,

u m = L l , m ( - f ) ,

f = (Tr 2 + x) sin rex,

m L~, o = 0 L~,o -- B f

1 2 3 4 5 6 7 8 9

10

0.493459 0.493360 0.490534 0.487425 0.486462 0.486180 0.485845 0.483459 0.483167 0.483087

0.001061 0.000994 0.000758 0.000541 0.000303 0.000121 0.000094 0.000090 0.000087 0.000081

Note tha t the solution of linear operator equations by the interpolat ion m e t h o d is advantageous in the sense tha t the realization of the interpolat ional process allows us to obtain an interpolant for the inverse operator, which is helpful in solving a wide class of problems.

Remark. If an in terpolant for A -1 is constructed at nodes fi , B f i = Aifi, ( f i , f j ) = aij, i , j = 1 , 2 , . . . , for the equat ion Au = f , then on the basis of (5.3) and (4.23), we have

]u* ~* ] < c ~ ( ~ m ~ ) - l / 2 l I f l l + , - - U r n y - - X+ = B I / 2 x C X .

w Application of operator interpolation to the problems of polynomial system identification First we consider the simplest cases of identification of linear and quadrat ic operators in a Hilbert

space X by the operator interpolat ion method. Let F: X --4 Y be a linear cont inuous operator , let {~i}i~l be an o r thonormal basis in X; and let c2i be eigenelements of the operator B corresponding to the eigenvalues Ai > 0, i = 1,2, . . . . To identify F(x) , consider a part icular case of the polynomial PI ,m(x ) in (2.4)

w i t h n = l , L 0 = L 1 = 0 E Y :

P~,m(x) = <]~,l- '~-'~(x)), (7.1)

2509

where

I'a = diag{~a,~2,. . . , ,X~}, I'~ -~ = diag{~7~,,~-~,...,,~7~l},

Taking into account (7.1)-(7.3), we write the operator P~,m in the form

i=1 i=1

Naturally, in this s i tuat ion the interpolat ional process converges in the metr ic of the space Y:

II II f[( )ll [[ -- I = fi~(X)- E F(~Pl)(X'~i)X -~- F 37- E(37,~Pi)x~i F(37) P~I'm (x) Y ,=, Y i=, Y

i = 1 X

Now let F(x) be a quadra t ic operator . Then we consider I ^ P~',m in (2.4) with n = 2, L0 = L~ = L2 = 0 ~ Y:

1 <-~, F;1/2(37)> ' (7.5) PI,m(X) -= -~.

where

~=t = 2! (F(B~,,B~ol),F(B~o~,B~o2),...,F(B~I,B~om),

F(B992 , Bop2 ), F(Bq~2, B~fl3 ) , . . . , F ( B ~ 2 , Bc?m ) , . . . , F(B~m_,, B~m ), F(Bcflm, B~m )) /

2l (.X21F(~l, ~91), /kl.~2F(~l, ~P2),. . . , .Xl/kmF(~l, ~m), ('/.6) \

/~2F(~2, ~2 ), ~2/~3-b-~(~2, ~3 ) , . . . , )~2)~m F(~2 , ~ m ) , . . - , /~m--1/~mF(~m-1, ~Pm), ,~2mF(~m, ~Pm)),

{ 1~ 1 1A 2 } (7.7) r~ = diag ~ , ~ ,,~, 7 ; ,1~3 , . , , ~ , . . , ~ ,,,-l~r,,,;',,, ,

F~ -1 = diag {/~12,2(,~1/~2)-1,2()~1,~3)-1,... ,)~22,... ,2(.~m_l/~m)-l,,~n2},

~(37) = ((x,~,)x(37,~,)x,(37,~,)x(37,~)x, . . , (37,~)x(x,~, ,)x, (37, ~2 )X (37, ~2 )X, (37, ~2 )X (37, ~3 )X, �9 �9 �9 , (37, ~2 )X (37, ~m )X, (7.8)

�9 "" ,(37,~m-l)X(37,~m)X,(37,~m)X(37,~m)X).

Taking into account (7.6)-(7.8), we represent operator (7.5) in the form

/9I 2,re(X) = F(Tl, ~l )(X, 991)X(X, ~l )X A- 2F(q~,, ~2)(x, ~a )x(x, ~2)x + 2F(q~,, ~3)(x, qol )x(x, qo3)x + ... + 2F(~1, ~m)(x, ~, )x(x, ~,,,)x + F(qo2, ~22)(x, ~2)x (x, qD2)x + 2F(T~, qv3)(x, qo2)x(x, Tz)x + " "

-4- 21:;'(qOm-l, q~m)(X, q~m-l )X(X, ~Pm)X -4- F(~pm,~Pm)(X, qOm)X(X, qOm)X

: F(~i ,~k)(x ,~i )X(X,q~k)X : F (x,~i)Xq~i, (X,~k)X~Pk �9 i,k=l -- i=1 k=l

(7.9)

2510

From the continuity of F and formula (7.9), the convergence of the interpolational process P~,m(x), as m --+ oo, follows.

It is easy to show that for the kth operator degree F ( x , x , . . . , x ) the interpolant PI,m(X ) can be

k written in the form

P l , m ( X ) = F ( x , ~ o i t ) x ~ i , , ( x , ~ i 2 ) x ~ i ~ , . . . , (x, qpik)x~oi , , i1=1 i2=1 i k = l

r t while for the operator polynomial P,~(x) = ~k=0 Lk(x) it can be written in the form

n ( m ,,~ m )

k=O i1=1 i2=1 i k = l

Since {~oi}~a=l is a basis in X and Lk(x) are continuous, from (7.10) the convergence of P~*,m(x) to P,~(x), as m --+ o% follows.

Comparing (7.10) with formulas (2.5.12) and (2.5.13), we arrive at the following conclusion [57]. If Pn(x) is a functional polynomial, the identification by (7.10) coincides with the identification carried out by the method of orthogonal moments [57]. However, as is not difficult to see, the solution of the identification problem obtained by the operator interpolation method (2.4) has an essential advantage over the solution- obtained by the method of orthogonal moments, namely: the method of orthogonal moments is based on the fact that in the polynomial model of the object to be identified the input signal x is represented in the form of a t runcated Fourier series. It is obvious that in this case such an approximation always converges to the mathemat ica l model (of a polynomial operator) as m --+ c~. But if the system {~Oi}ic~_ 1 is not a basis, then the approximation under consideration does not, in general, converge, and it is extremely difficult to get estimates of accuracy. As is shown in this chapter (Theorems 4.2-4.5), the operator interpolation method ensures extremely important properties of the interpolation error in the metric of the space H,(Y) , and in the case of a complete system {~i}ie~ 1 i t e n s u r e s also the convergence of the interpolation process in this metric.

w D e t e r m i n a t i o n of t h e t r a j e c t o r y of an o b j e c t by t h e p o l y n o m i a l o p e r a t o r i n t e r p o l a t i o n method

Statement of the problem. The trajectory of an object is sought as the curve of intersection of two surfaces in the space JR3. Using the Weierstrass theorem generalized by Stone to the class of continuous functions from Nn [132], we represent the trajectory in the form of the intersection of two nth-order surfaces:

{ V . l O , ) = k = o .

iWjWk<n

= b jkX yiz k = O, i+j+k<_n

aijk, bijlr E I~1. ( 8 .1 )

We consider the object as a source of radiation of a periodic signal at a certain frequency. Then, using the phase method proposed by the author of [82] for the case of a spherical wave front, at moments t l , t 2 , . . . , t m we find coordinates of points ui = (xi,yi,zi), i = 1 , . . . , m , of the object and then, with the aid of polynomial operator interpolation, we choose the coefficients aijk, bijk so that the curve line (8.1) passes through the points ui, i = 1 , . . . , rn. In what follows, computational errors are not taken into account.

Thus, we divide the problem of finding the object trajectory into two stages. At the first stage the coordinates of points vi, i = 1 , . . . , m, of the object are determined by the phase method [82], while at the second s t a g e t h e nth-order surfaces in (8.1) passing through points ui, i = 1 , . . . , m, are constructed by the operator polynomial interpolation.

Thus, let the Cartesian coordinate system O X Y Z be given, and let NI and N2 points that are the points of location of measurers (the origin of coordinates is the point where the reference measurer is

2511

located) be given on the rays OX and OY, respectively. The measurers serve for obtaining the signal's phase information, to be described in detail below. With measurers located on straight lines, the phase method proposed by one of the authors in [82] permits one to determine, at moment t, the distance between the reference measurer and the object itself and the direction cosine of the radius vector of the object's location. Thus, by means of measuring systems located on the rays OX and OY, the problem of determining the coordinates of the moving object at a fixed moment t is solved.

Consider in more detail an algorithm for the phase method [82] in the case where the measurers are located on the ray OX. Let 62i be the total phase difference between signals received by the reference and the ith measurers, let R be the distance between the reference measurer and the object itself at moment t; and let di be the distance between the ith and the reference measurers, 0 < dl < d2 < . . . < dN~. All the quantities are reduced to the wavelength of the signal radiated. Denote by a the angle between the radius vector of the object 's location at moment t and the axis OX. Then, on the basis of the results of [81, 82], in the case of a spherical wave front, the mathematical model for determining R and cos a from the phase-metric information contained in a periodic signal has the form

(I)l -I- c (d 2 - (I)2) 62m ~- E(d2m - 622) (8.2) z = d l = " ' " = d m '

e = 1/2R, z = cos a.

In the model (8.2), the measuring system consists of m + 1 measurers, m <_ N1. Despite the imaginary simplicity of the system of equations (8.2) for determining R and z, its solution evokes great difficulties. This can be explained by the fact that with the help of a pair of measurers (the ith and the reference ones) at moment t we obtain not the value of the total phase 62i but the value of its fractional part q0i = {62i}+. Therefore (8.2) is a system of m - 1 equations with respect to m + 1 unknowns k~, k 2 , . . . , kin, R (ki = [62~]+ is the whole part of the phase ffi), where di and cpi, i = 1 , . . . , m, are given. So, (8.2) is, actually, a nonlinear system of ra - 1 equations in m + 1 unknowns, among which m unknowns are integer. Moreover, as was shown in [82], in general the solution of problem (8.2) is nonunique. However, it .is possible to indicate intervals (el, f l i ) such that for di E ( o t i , f l i ) , i = 1 , . . . , m, system of equations (8.2) has a unique solution. In other words, from N1 measurers it is necessary to choose m measurers so that the above conditions be fulfilled. Let us dwell in more detail on the algorithm [81, 82], taking into account the assumption that the following restrictions are fulfilled: dl < 1, z > edm, R E [R1, R2], where R1 and R2 are known.

Without loss of generality, we present this algorithm for the case of three measurers (m = 2). From system (8.2) we get

622 = R - - ~-D--22,

D2 = R 2 + d~ - h1(2R621 + d 2 - 62~), d2

hi ~-- ~1-1' 621 = ~1, (8.3)

since k, = [6211 + = 0 (dl <_ 1, ~ - 6212 > 0). For given d,, ~ , , and d2, the function 622 in relation (8.3) is a function of R and 622 = 622(R), R E [R1, R2]. Obviously, given ~02, the problem of determining k2 = [622] + and coordinates (R, a) has, in general, a nonunique solution. Let us investigate the function 622(R) for R E [R1,R2], R1 r R2, and given dl, ~1, d2. The case of R1 = R2 is trivial since R is known and k2 = [622] + is uniquely determined from relation (8.3). It is not difficult to see that on the segment [R1, R2] the function 622(R) increases monotonically in R (0622/0R > 0). Put A622 = 622(R2) - 622(R1). Since the function 622(R) increases on [R~, R2], the analysis of relation (8.3) allows us to obtain the following result.

1. For A622 > 2, system (8.2) always has a nonunique solution for any ~2. 2. For 1 _< A622 < 2, system (8.2) can possess both a unique and nonunique solution depending on ~02

and the number of integer points in the range of variation of 622. Thus, if

5622 = [622(R2)] + - [622(R,)] + = 1,

then the parameter k2 is uniquely defined under the condition qo2 E ~2, where

= I e M u N } ,

M = {q~2 J ~2 < {62~(R1)}+ }, Y = {~2 I c22 > {622(R2)}+}.

2512

Here

k~ = { [*~(R~)]+'r, ~1 ~ 6 M,N. p~ ,R , , +, ~2 e

The parameter k2 is not uniquely defined for ~2 r ~2. If 6~52 = 2, then the parameter is unique under the condition

Here ~ : [r + + 1 : [r + - ~.

If ~02 ~ ~2, then k2 is not uniquely defined and system (8.2) has a nonunique solution. 3. Let Aq~2 < 1. Then system of equations (8.2) always has a unique solution for any qP2. In the case

692 = 0 we have

k2 : [O2(R1)] + : [~52(R2)] +, (8.4)

while in the case 6ff2 : 1 we have

{ [r +, ~ >_ {r + k2 : [O2(R1)] +, ~2 <_ {~2(R2~)} +' (8.5)

Let us show, for example, how to obtain the above result for ACe < 1. The following cases are possible: if 6~2 : 0, then n + el _< ~2 _ n + e2, 0 < 61 ( e2 ( 1; if 6ff2 : 1, then l + el _< if2 <_ l + 1 + e2, 0 < e2 _< el < 1, where n, l > 0 are integers. Since the true value of R is located on the segment [R1,R2] and the function q~2(R) increases in variable R, in the first case the parameter k2 is defined uniquely for d2 and given ~P2 6 [el,e2]. Here

ks : [r : [*~(R1)] + [r (n)]+ = 2 2 =7%-

In the second case, in view of the monotonicity of g22(R), either the inequality

~ _> e, : {r + or ~ _< e2 : {r +,

or the inequality (8.5) holds, and the validity of (8.5) and hence the uniqueness of parameter k2 is obvious. In the cases 1 _< Aft 2 < 2 and Aft2 > 2, the results are derived in the same manner.

Thus, for d2 > dl and corresponding T2 we arrive at one of the three cases considered above. If k2 is determined uniquely, then having evaluated it, we find R from Eq. (8.2) (this value of R is denoted by R(2)) by the formula

l h~(4 - ~) - (~ - ~) R(2) : 2 ~] : h - , E (s.6)

Knowing the values of k2 and R(2), we find

2 m 2 ) r + 4 - r a(2) : arccos 2R(2)d2 (8.7)

However, the arbi t rary choice of d2 > dl (the arbitrary choice of the third measurer) does not always lead to the uniqueness of parameter k2 and, respectively, to the uniqueness of the solution of system (8.2). The examination of various values of d2 (i.e., of various measurers from those available) with the aim of obtaining the uniqueness in the evaluation of k2 is inadvisable for many natural reasons. Therefore, the problem arises of finding the set 7, of values of d2, for each of which the system of equations (8.2) (m = 2) has a unique solution. For given dl < 1, ~01, R1, R2, the set Z can be chosen to be the set of solutions of the inequality Aq~ 2 < 1.

2513

Taking into account (8.3), we have

D2~-~2) - V / ~ R I ) > AR - 1, AR = R2 - RI. (8.8)

A solution of inequality (8.8) can be obtained analytically, but in view of the cumbersomeness of the formulas we do not present it. Thus, let Z be a set of solutions of inequality (8.8). Then, choosing d2 E 7, (respectively the third measurer from N~ measurers available), we evaluate k2 by formulas (8.4), (8.5), while the values of R(2) and 0~(2) are evaluated by formulas (8.6) and (8.7). However, in practice the quantities ~i = {~i + r are measured, where r are random errors. Therefore, for correcting the values of R and c~, the number of measurers is increased. Let, for example, ki and, respectively, R(i) and a(i) be found according to the ith scale. It is required to correct the values of R and a according to the (i + 1)th scale, i.e., it is required to find R(i + 1) and ~(i + 1). From system of equations (8.2) we have

f f i + l ( R ) = R - V ~ / + I ,

"Di+I = R ? + d2+l - 2Rdi+l cos o~(i).

For fixed di+1 and cos a(i) the function ~i+1 (R) is a monotonically increasing function in the variable R (0Oi+I(R)/0R > 0), and the results obtained for the case m = 2 are valid for arbitrary m. For example, the set of solutions of the inequality

V / ~ ) i + I ( R 2 ) - v/~Di+I(R1) > A R - 1, i _> 2,

can be chosen as the set Z of values of di+l (i _> 2), for which the parameter ki+a (R(i + 1) and c~(i + 1) respectively) is determined uniquely, while the formulas for determining R(i + 1) and o~(i + 1) according to the (i + 1)th scale have the form

1 hi(d~i (I)2) __ (d~/+l 2 - - (I}i't-1) hi - -

R(i + 1) = 2 (I)i+ 1 -- hiff2 i '

2 2 2R(i + 1)ff2i+l + di+ 1 - ~i+1 c~(i + 1) = arccos

2R(i + 1)di+1

di+l d/ ' (8 .9)

(8.10)

Thus, we define the values c~ = c~(m) and R = R(m) by formulas (8.9) and (8.10) with i = m - 1. With the help of the system of measurers located on OY, in an analogous way we define the angle/3 between the radius vector of the object's location at moment t and axis O Y as well as the distance between the reference measurer and the object itself. As a result,

Rcosol, Rcos/3, and R v / 1 - c o s 2 a - c o s 2 / 3

will be the coordinates of the moving object's location at moment t.

Remark. In [82, 83], an accuracy analysis is carried out for the estimates of the object 's coordinates obtained by the phase-measurement method for a spherical wave front. In [80], an efficient method of phase measurements is proposed for solving the problem for the flat-parallel wave front.

Thus, let coordinates of points vi = (xi,yi, zi), i = 1,. . . ,rn, of the moving object be determined at the moment ti by the phase-measurement method considered above. Now let us turn to the second stage of solving the main problem of the section, namely: with the help of the polynomial operator interpolation we define the coefficients ~ijk and/3ijk in (8.1) so that the trajectory of the object 's movement in (8.1) passes through the points vi, i = 1 , . . . , m. Note that if M is the number of coefficients in the nth surface equation, the fulfillment of the condition m >_ 2M is sufficient for system of equations (8.1) to determine a unique curve line.

Consider the polynomial operator interpolant pI in (4.16) of Chapter 2, where X is a Hilbert space,

: (v, 17 = # e y m

2514

Y is a vector space, V - = V +, and V + is the Moore-Penrose matrix pseudoinverse to the matrix

n ~j=l " v = p=O

Then pI(v) acquires the form

P~(v) = Qn(v) - "' V+ E (v, vj)~ , p=0 j=l

(s.11)

where Q, E 1-I,. Operator polynomial (8.11) interpolates the operator F, which takes zero values from Y at the nodes vi,

i = 1 , . . . , m. Such an interpol0nt always exists because the corresponding interpolation problem is always solvable: ZvP 2r = ~ (Theorem 4.6 from Chapter 2). Now let

X = R 3 , Y = R 2 , v i = ( x i , y i , z i ) , i = 1 , . . . , m ,

where (xi, yi, zi) are coordinates of points of the moving objects's location obtained by the phase-measurement method [82] at moments ti, i = 1 , . . . , m , and ( . , . ) x = ( ' , ' ) is the scalar product in R3. Put

Q.(v) = (Qnl(V),Qn2(v)),

~ n ( v ) = (Q,(v l ) ,Q,(v2) , . . . ,Qn(vm)) , m

(~ = EO~ifli ' Oli e R2, ~i e R1. i=1

Then, by virtue of (8.11), the equation of the trajectory of the object's movement as the intersection line of two nth-order surfaces is written in the form

n

Pil (v) Qnl(v) ( " ~ n l ' u + E {(V, p m N) = - v i ) }i=, =o, p=0 /

p m

p=O 2

Q.1 # 0 . 2 , Q . iCKerpI i (Q) .

(s.12)

Thus, by the polynomial operator interpolation method, an explicit analytical representation of the object's movement trajectory equation is obtained in the form (8.12) as the intersection line of two nth-order surfaces, without solving the corresponding system of linear algebraic equations.

Thus, in the simplest case, n = 1, rn = 2, Vl = (0,0,0), v2 = (1, 1, 1), by setting, for example,

Qll(V) : 2 - x + y + z , Q12(v) = 2 + x + y + z ,

we have

111111 v+=v-i V = 1 4 '

1

E {(v'vi)'},2=, = (1,1 + x + y + z) T, p=O

1bl4 1tl 3 -I 1 '

~]11 = ~]12 = (2,3) T.

2515

In the case under consideration, the system of Eqs. (8.12) is equivalent to the system

2 x - y - z = O ,

x + y - 2 z = O ,

which determines the movement trajectory as the intersection line of two planes. Note that if we put Qll( t ' ) = 1 + x + y + z and Q12(v) = 1 - x - y - z in formulas (8.12), then

the left-hand sides of Eqs. (8.12) are identically zeros, and the straight line cannot be determined as the intersection line of two surfaces. This can be explained by the fact that Q ~ ( v ) and Q12 (v) so chosen belong to the kernels of operators P~I (QI) and P~2(Q1), respectively. Hence, for determining some trajectory as the intersection line of two nth-order surfaces with the help of system of equations (8.12), it is necessary to require that Q,~i r K e r p I i ( Q n ) , i : 1,2.

Let us formulate necessary and sufficient conditions for the polynomial Q , E l-I,, to belong to the kernel of the operator P~(Qn) in (8.11) for the case of an abstract Hilbert space.

T h e o r e m 8.1. In order that Qn E KerP~, P~: I I , --+ Ha, it is necessary and suff icient that the representation of Q,~ in the f o rm

Q.(~) = ~ ,~ (v , v , ) '~ i=1 p=0

exist, where c~i, i = 1 , . . . , m , are arbitrary elements f rom Y .

Proof. Necessity. Let P~(Q, ) = 0. Then, setting ~,~ = V~7, ~7 = (cq, . . . ,o~m) r , and taking into account Lemma 4.5 from Chapter 2, we have

= v/)x},=l} p=O

__ c~,VV+E{(v ' p m - -" v i b } , = l p=0

= <Y(~, Y + L {(U, Vi)~ }1%1) p=O

< L > p m = S , ( E - ~ o ) {(v,v,)~},__~ p=0

m I%

= E O Z i E ( V , vi)Px �9 i=1 p=0

Sufficiency. Let

O,~(v) = Oti E ( V , vi)P X i=1 p=0

and let gi be a vector from Rm whose i th coordinate is equal to t and the rest of which are zeros. Then the expression in ( - ) from (8.11) acquires the form

W + {(v, P m . V_ t_ v~lx}~=, ~ (~,,~) (~,v~) 3 ' - - J i - = l l p=0 i=1 "=

,=1

<< n >

~ V+ E p m {(v,v,b},=a p=0

~ p r n ,~,e,, v v + ~ {(~,~,)x},=l p=O

p----0

"~ i=1

~,~,, {(v,~,)x},__, = ~ , ~ ( v , v , ) ~ i----1 p=O i=, p----O

From here, on the basis of (8.11), we have P ~ ( Q , ) = 0, which completes the proof of the theorem. []

2516

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Polynomial interpolation of operators