Math. Nachr. 180 (1996), 289-303

Simultaneous Approximation by Polynomial Projection Operators

I3y T. F. XIE of Hangzhou, and S. P. ZHOU of Hangzhou

(Received November 2, 1993) (Revised Version December 22, 1994)

Abstract . Pointwise estimates are obtained for simultaneous approximation of a function f and i ts derivatives by means of an arbitrary sequence of bounded projection operators with some extra condition (1.3) (we do not require the operators to be linear) which map C I - ~ , ~ ~ into polynomials of degree n, augmented by the interpolation off at some points near f 1. The p w e n t result essentially improved those in [BaKi3], and several applications are discussed in Section 4.

1. Introduction

Let C ~ - l , l l be the class of functions which have q continuous derivatives on [-1, 11, and let c > 0 be a real number. For each n E N define

Yn =: { - 1 = t o 5 t l 5 ... 5 tr-l 5 -1 + c/n2,

1 - c /n2 5 t r 5 tr+l 5 * * 5 t2r-1 = I}, (1.1)

(1.2) y; =: { -1 = t o = t l = ... = ti , < til+l 5 * * * 5 tr-1 5 -1 + c/n2,

1 - c/n2 5 t r 5 * * . 5 t2r- l1-2 < t 2 r - / 2 - 1 = ... = t z r - l = 1).

In recent years, simultaneous approximation to differentiable functions and their derivatives by interpolating polynomials has been investigated extensively (see [BaKil], [BaKia], [Mun], [RuVe], [Sza], [XiZhl], [XiZh2], etc.). By a motivation like this, [BaKi3] recently invesgated projection operators which are more general than interpolation polynomials. Write

1991 Moihcmaiics Subjcci Claarificaiion. 41A05 41A10 41A28 41A35. Keywords and phnascs. Simultaneous approximation, polynomial projection operator.

290 Math. Nachr. 180 (199(i)

and let Ln be a bounded linear projection which, if q is even, maps even 27r - periodic, functions to cosine polynomials of degree n or, if q is odd, maps odd functions to siiic,

polynomials of degree n. For gn E Cf-,,,] such that gn/Qr,n E C[-1,11, define

With all these notations, [BaKiJ] proved that:

“Let gn E C~-l,ll be such that gn/Or,,, E C[-l,li, and T = [q]. Then the followi~ig estimates

lghk)(z) - Tn+q(gniz) (k) I 5 ~ A i - ~ ( z ) En(gn (Q) 11 L rill hold uniformly on [-I, 11 for 0 5 k 5 q, where En(f) is the best approximation of f E C[-1,11 by polynomials of degree n, II.II = max 1 .I, and IlLll =: sup IIL(f)ll.”

-1<2<1

In the above theorem, when q is odd, rn+q does not need to be an algebraic polyno- 11111=1

mial. As an alternative, [BaKi3] established that for an odd number q,

where Z:(f,z) = J T T ? L n ( f ( t ) / J m , z ) .

Under what conditions can we replace An(z) by 6n(z) =: d i / n ?

ators which map nomial into itself) and which satisfy the following condition

There is therefore an important natural question arising from the above results:

For f E Cf-l,ll, let {Ln(f,z)} denote a given sequence of bounded projection oper- into the polynomials of degree n (and map any n - th poly-

for any f E C[-l,ll and any polynomial p of degree n (note that we do not require L,, to be linear. If Ln is linear, it obviously satisfies the above equality). For gn E Cp-, such that gn/Qr,n E C[-l,l], define

Qr, n

where here and in what follows, or,,,(.) is defined on Y,’ (see (1.2)) as above.

follows. Some important applications will be given in Section 4. The aim of the present paper is to give a complete answer to the above question iih

where R z , . - ~ , ~ ( f,x) is the Lagrange interpolating polynomial o f f of degree 2r - 1 on Y,' .

where

and

Remark. It is clear that llLAll* 5 [ILAIl, we make this slight variation to fit in more convenient range of applications (see Theorem 3 in Section 3). In general, the approximation rates in the above two theorems cannot be improved (see [XiZhS]).

292 Math. Nachr. 180 (199(i)

2. Lemmas

In Lemmas 2.1 -2.3 and Lemma 3.1, we assume that q is a nonnegative integer. The following theorem can be found in [Kill.

Lemma 2.1. Let f E C~- l , l l . Then there are polynomials P N ( Z ) of degree N (greater than or equal to 2q) such that for 0 5 k 5 q,

lf( ' )(~) -P$) (x ) l 5 M J k - - " ( s ) E N - q ( f ( ' ) ) i 1x1 5 1 1

where M indicates a positive constant only depending upon q, and &(x) = Moreover, j ( q ) ( * 1) = p p ) ( j z 1).

We establish the following lemmas. We need the following Lemma 2.2 for our purpose. However, we prove the following

Lemma 2.2' first, which is a more suitable form of this independent result.

Lemma 2.2'. Let c > 0 be a given real number, Y,' be the system of nodes (1.2), 1 5 T I (q + 2)/2, and 1 = min { 1 1 , 1 2 } . Assume that f E Cr-l,lI, then there i s ( I

polynomial Pn(Z) of degree n (greater than or equal to max {2r, f i } ) such that P,(x) interpolates f (x) on Y,' when Y,' has coalescing points, for example, t j = t j+l =

.. . = tj+1, then P?)(t,) = f ( i ) ( t j ) , i = 0,1, . .. , I (

and for -1 5 x 5 1, )

d m where 6,(z) = n

Proof . From the Gopengawz' theorem (see [Gop)), we knew that for f E C!-,,,,, there are polynomials pn(x) of degree n such that

(2.2) If'*'(%) - p i k ) ( x ) I I ~ J ~ - * ( ~ ) W ( f ( ' ) 1 6 n ( z ) ) r 121 I 1 ' 0 5 k I q

Take p,(x) := 41, (x) as the polynomial satisfying (2.2), which implies that qI1 (2) is ii

Lagrange interpolating polynomial of degree n of f on

{ -1 = to = . . . = ti , , t2r-12-1 = * * * = t2r-1 = 1)

(since I l l l2 I T-1 I q/2). Without loss of generality, assume that 11 5 12. Write qm(x ) as the Lagrange interpolating polynomial of f - q j ( x ) of degree m + 12 + 1 oii

Y,' \{tm+1, tm+2, . . . , t~r-12-2) when 11 < m 5 1 2 , and qm(z) as the Lagrange interpcp lating polynomial of f - ~ ~ ~ l l qj(z) of degree 2 m + l on Y:\{tm+i, tm+2, . . . , t2r-m-2 I

Xie/Zhou, Simultaneous Approximation 293

when l2 < m 5 T - 1. We will prove that P,(z) =: Ekzll qm(z) is the required poly- ~romial. Notice that p,(~) is clearly the Lagrange interpolating polynomial of f of degree n on Y,'. To prove that P,(z) satisfies (2.1), we will establish the following inequalities by induction:

For every 11 + 1 5 m 5 r - 1, and z E [-1, 11,

p%)l ( ~ ) ~ ( f ( 9 ) , 6 n ( ~ ) ) , 0 5 k L 1 ,

1 + 1 5 k L q ,

21+k-1 (x) 6;1-2k+1

M A ~ - , - " ( z ) w ( ~ ( ' ) , An(z))i

(2.3)

and, consequently, by (2.2) and (2.3), for every 11 5 m 5 r - 1 and 3: E [-1, I],

MAz-21+k-1(z) 6:1-2k+l (z>w(f'9',6n(Z)), 0 L k 5 1 , { M A ~ - k ( ~ ) ~ ( f ( q ) , ~ n ( ~ ) ) , 1 + 1 5 k 5 q .

For convenience, we only discuss here the case that Y:\{l,-l} has no coalescing nodes, the other case could be treated similarly without any difficulty.

Due to Gopengawz' theorem, (2.4) clearly holds for rn = I I . Suppose that (2.3) as well as (2.4) holds for rn, 11 5 m 5 r - 2. Assume first 0 5 k 5 1. We divide the proof into the following several cases.

Case 1. m < 12 and z E [-1,0]. In this w e , t m + l E [-1, -1 + c/n2]. Write

m

x 1 (Qm(z) (1 + z)'I+l (1 - z ) ~ ~ " ) ' ~ ) I.

294 Math. Nachr. 180 (199fi)

has at least m + 2 zeros tj, j = 0,1, . . . , m + 1, in [-1, -1 + s], consequently thew is a <1 E [-l,tm+l] C [-1, -1 + +] such that

f (m+l) (<I) - 2 q:"+"(<l) - Al(m + l)! = 0 , j=l1

or equivalently, by induction hypothesis,

Therefore it follows that

At the same time,

X ialZhou, Simultaneous Approximation 295

in view of that q 2 2r - 2, m 5 r - 2 and thus 2q - 4 m - 4 2 0, that is, (2.3) holds in this case. Obviously (2.4) follows by (2.3) and induction hypothesis. We have finished Case 1.

Case 2. m < 12 and z E (0,1]. In this case, tm+1 E [ - 1, -1 + c/n2] . Again write Qm(z) = ~ ~ = , , + , ( z - t i ) , with the same argument as Case 1 we have estimates (2.5) and (2.6). And similarly,

Combining (2.5), (2.6), (2.9), and (2.10) yields

since m 5 12 - 1, 12 5 r - 1 and thus 2q - 412 2 0, therefore (2.3) holds in this case.

296 Math. Nachr. 180 (l!l!h)

Case 3. 12 5 m 5 T - 2 and z E [-1,0]. In this case, tm+l E [ - 1, -1 + c/n2] iiliil

t2r-m-2 E [l - c/n2, 11. Now let

i=l1+1 j=2r-m-1

Because qm+l(z) interpolates f(z) - C,”=ll qj(z) on Yn \ {tm+29 . . . , t2r-m-3},

(2.11)

At the same time, write

has at least m + 2 zeros t2r-m-zr . . . (2 E [t2r-m-29 1) C [l - 3,1] such that

in [l - 3,1], and thus there is i i

m

f(”+’)((2) - C qjm+”((2) - Az(m + l)! = 0, j=[1

Xie/Zhou, Simultaneous Approximation 297

or by induction,

Thus

Similar to Case 1, we estimate

(2.14)

298 Math. Nachr. 180 (19!#(i)

Together with (2.11) and (2.12)’ (2.3) as well as (2.4) holds. Case 4. 12 5 m 5 T - 2 and z E (0’11. The argument is the same,as Case 3. From all the above discussions we have proved Lemma 2.2’ for 0 5 k 5 1. Now for I1 + 1 5 rn 5 r - 1 and x E [-1’11,

lq:)(x)l 5 ~ n ( z ) )

by (2.3)’ applying Bernstein type inequality (see [Lor]) we get for 1 + 1 5 k 5 q that.

lqg)(x)l 5 MA:-^(^) w(f‘q), An(z)).

Therefore Lemma 2.2’ is completed. a Using KILGORE’S result (Lemma 2.1) instead of Gopengawz’s theorem, we obtain

similarly that

Lemma 2.2. Let c > 0 be a given r a l number, Y,’ be the system of nodes (1.2), 1 I r 5 (q + 2)/2, and 1 = min {11,12}. Assume that f E Cf-l,ll, then there is (1

polylomial Fn(z) of degree n (grater than or equal to max {2r, &}) such that Fn(x) interpolates f(z) on Y,’ and for -1 5 x 5 1,

(2.15) MAt-2I+k-2 (x) 6:1-2k+2 ( ~ ) ~ n - - 9 ( f ( q ) ) 1 0 5 k I 1 , 1 + 1 5 k 5 q . ‘ ( M A z - k ( ~ ) En-9 (f‘4’)’

Lemma 2.3. Let Fn(z) be as defined in Lemma 2.2. Then for x E [-1’11 and an even q,

Lemma 2.3’. Let pn(x) be defined as in Lemma 2.2. Then for x E [-1’11 and un odd q ,

The proofs of Lemmas 2.3 and 2.3’ are similar to those in [BaKi2].

Lemma 2.4. Let c > 0 is a given recrl number, Y, be the system of nodes (1.1) given in Section 1, and T = [ (q + 1)/2]. Assume that f E Cf-l,ll, then thew is polynomial pn(z) of degree n + q (greater than or equal to max {2r, &}) such that Fn(z) interpolates f(s) on Yn and for x E [-1,1],

If(’)(z) - FAk)(x)l 5 MA~-,-k(x)En(f(9)), 0 5 k 5 q .

X ie/Zhou, Simultaneous Approximbtion

Proof . The lemma is a theorem proved in (BKV].

299

0

3. Proofs

P r o o f o f Theorem 1.1. We suppose that 0 5 k 5 1 and d m 5 1.1 5 1, - 1 5 3: 5 - @ and for the other case the argument is similar. Take the polynomial

pn(z) in Lemma 2.2 which satisfies (2.15) for f = gn, and we see that

Write

i = l r + l

Since Fn(z) interpolates f on Y,', we then have

by the definition of Pn+q. F'rom Lemma 2.3, we get

(3.3)

Obviously, from the assumption of Y;, for 0 < i 5 k - j 5 11,

( Q t ' ( x ) l

(3:) 9

0 5 i 5 T - I l - 1 ,

M , r - 1 1 5 i 5 k - j ,

2r-211-2i-2 < Mn2r-21 1-2 i -2AF-211-2i -2 M ( d i = 7 + l / n ) -

and

300 Math. Nachr. 180 (l!l!)(i)

Then

I = &(;) j=O

x L?]

k

By Bernstein type inequality (see, for example, [Lor]), for any given 5 E [-1, 11,

With (3.2), (3.3)

k

n ( g i g ) ) IILII C n 2 j 6 i ( , ) I < ~ ~ - q + 2 r - 2 k 2r-211-2( ,4 J : 1 1 - 2 k + 2 ~

- An j=O

5 ~ ~ ; - 2 " + k - 2 ( 2 ) p - 2 k + + 2 ~ , ( g i q ) ) l l ~ ~ l l

in view of -1 5 x 5 -,/- and thus n26,(x) 5 1 as well as An(2)/2 5 n-2 5

Altogether with (2.15), (3.1 , by applying Lemma 2.2, we have obtained the desired result for 0 5 k 5 1 and s- 1 - l / n 2 5 Izl 5 1.

In the case 0 5 k 5 1 and Izl 5 ,/-, or in the case 1 + 1 5 k 5 q, the required result of Theorem 1.1 is concluded from the following proposition, which we write its a lemma:

An (z) *

Lemma 3.1. Let q be an even integer, gn E Cf-l , l l be such that gn/Qr,n E C[-l ,I] , and T = [q] . Then for k = 0,1, . . . , q, the following estimates hold uniformly on I-1, l] :

Igkk) ( s ) 7 pz)q(gn,z)( I MA:-'(^) En ( 9 2 ) ) IILII.

S io/Zhou, Simultaneous Approximation 301

PIoof . The proof is almost the same as above, and even much simpler, by using I,oirima 2.4 instead of Lemma 2.2. Actually, in a similar way we can prove that

I M A x ( x ) E n (d9)) llLn11.

Applying Dzyadyk's inequality (see, for example, [BaKiP, p.231)) we get

Ipz9 ( g n - Fn, x) I I M w ~ ( ~ ) En (g?) ) IlLnll

f o r 0 5 k 5 q. Finally applying Lemma 2.4 into (2.15) (using Fn(x) to replace Fn(x))

P r 00 f of Theorem 1.2. The proof is similar to that of Theorem 1.1 by using Lemma 0

yields Lemma 3.1 0

2.3' instead of Lemma 2.3.

4. Applications

From Theorem 1.1 we deduce a particular case to be written as the following theorems which improve the results in [BaKil, 21.

Theorem 4.1. Let q be an even number, f E Cy-l,ll, T = [ (q + 1)/2], and 1 = rnin { 1 1 , 1 2 } . Then the following estimates hold uniformly on [-1,1] :

MAi-21+k-2(x) 6:1-2k+2 ( x ) En(f(q))IILnII, 0 I k I 1 ,

1 + 1 5 k I q ,

where Pn( f , x ) is the Lagrange interpolating polynomial of degree n + 2~ - 1 of f on the nodes {xcj : -1 < xo < X I < . . * < Zn < 1) UY,'.

Theorem 4.2. Let q be an odd number, f E C~-,,,l, T = [ (q + 1)/2], and 1 = min { 1 1 , 1 2 } . Then the following estimates hold uniformly on [-1,1] :

MAi-21+k-2(Z) 6:1-2k+2 (.)~n(f(Q))IIL;II*, 0 I k I 1,

Z+1 I k I q ,

where Pn( f, x ) is the Lagmnge interpolating polynomial of degree n + 2r - 1 of f on the nodes { x j : -1 < 20 < x1 < < zn < 1) UY,'.

302 Math. Nachr. 180 (19Yti)

Another corollary is the following

Theorem 4.3. Let f E C~Tl,ll, r = [q], and 1 = min{11,12}. Write Pn(f,z) lo be the best polynomial appronmant of degree n o f f , then the following estimates hold uniformly on [-1, 13 :

where R2r-l,n(f,x) is the Lagmnge interpolating polynomial off of degree 2r - 1 071

Y,’ . Proof . We see that {pn( f, z)} is clearly a sequence of bounded projection operators

satisfying (1.3). It is also not difficult to see that the modified norm for the besl 0 approximation operator, lip: 11’ , is bounded.

Remark. Note that the simultaneous approximation by best polynomial approxi- mants need not always be convergent (see [Zho], other related materials can be accessed in [Has] and [Lev]), the above result does supply an interesting alternative in this area.

Acknowledgements

The second named author was supported in part by Zhejaang Provincial Natuml Scienu Foundation of China and a Special Research f i n d of State Council of Educntion of China.

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(Jrina Institute of Metrology Department of Mathematics Ilangxhou, Zhepang 310034 Hangthou University (,'hzna Hangthou, Zhejiang 91 0028

China