Polynomially positive definite sequences

Math. Ann. 259, 487-495 (1982) Itlthemttsetv A m �9 Springer-Verlag 1982

Polynomially Positive Definite Sequences

Christian Berg 1'* and P. H. Maserick 2

1 Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark 2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Introduction

Using the shift operator E on sequences s=(s,),=>o, Stieltjes solution of the moment problem bearing his name can be formulated as follows : A sequence s is a moment sequence of a positive measure supported by [0, ~ [ if and only if s and Es are positive definite. Results by respectively Haviland [7], Svecov [12], and Devinatz [-5] can be stated that s is a moment sequence of a positive measure supported by respectively [ - 1, 1], IR~]~,/~[ and [-c~,/~], where c~ </~, if and only if s and respectively s-EZs, EZs-(~+~)Es+~s and fis-Es, Es-~s are positive definite.

For a polynomial p(x)= ~akx k with real coefficients p(E) denotes the shift operator p(E)= ~ akE k, where E ~ 1 is the identity operator, i.e.

(p(E)s),= ~ aks,+ k, n>O. (1) k

The above results suggest that one should examine the possible equivalence of the statements :

(i) s, pl(E)s,..., pl(E)s are positive definite sequences. (ii) s is a moment sequence of a positive measure supported by

K= {pl >O}n...n{pz>=O}. In the above statements Pl . . . . ,Pz are real polynomials. Section 1 below

contains mostly well-known material necessary in the sequel. In Sect. 2 we consider the case of l= 1 and characterize the polynomials P=Pl for which (i) and (ii) are equivalent, and we obtain in this way a result subsuming the results of Hamburger, Stieltjes, Haviland, Svecov, and Devinatz. In Sect. 3 we consider the case of arbitrary l and prove that if K is compact then (i) and (ii) are equivalent. We finally offer a few remarks about possible extensions of the above results to higher dimensions.

* The present research was carried out while the first author was visiting Pennsylvania State University with support from the Danish Natural Science Research Council, Grant 11-1648

0025-5831/82/0259/0487/$01.80

488 Ch. Berg and P. H. Maserick

1. Preliminaries

Let ~ = o~(IN o, IR) denote the vector space of real sequences s = (Sn)n=> 0 and let d denote the vector space of polynomials with real coefficients. The expression

(s, p> = Y'. skak, (2)

where

defines a non-degenerate bilinear form on f f x ~r under which ,~ and d form a dual pair.

We always equip d with its finest locally convex topology, which is easily seen to be the strict inductive limit topology of the finite dimensional subspaces de , d>=0, carrying their canonical topology, where ~r denotes the subspace of polynomials of degree <d. The topological dual space of d is equal to the algebraic dual space of d and can be identified with f f via (2).

For a closed subset K ~IR we denote by ~r the set of polynomials p~ d such that p(x)>=O for xeK. For K = I R we put d + = d +.

For a convex cone C ~ ' the dual cone C • is by definition

C ~ = { s e ~ l (s,p)_->0 Vpe C}.

By a theorem of Haviland, cf. [6, 7], the dual cone + • + (~ '~) of ~r is equal to the set of moment sequences of positive measures supported by K, i.e. the set of sequences s of the form

s.= ~x"d#(x), n>O,#~d4*(K), K

where d/*(K) is the set of positive Borel measures on IR having moments of all orders and support contained in K. If K is compact, any positive Borel measure on IR with support in K and #(K) < 0o belongs of course to J /*(K).

A sequence s is called positive definite if

si+jqc >=O i , j=O

for all finite sets {Co, q , ..., c,} _c R, or equivalently if (s, p2) > 0 for every p e sO. The set of positive definite sequences is denoted 9 ~. As is well known, every polynomial in ~r + is of the form p2 + p~ with p~, P2 E ,~r and it follows that s is positive definite if and only if se(~r • which by Haviland's theorem holds if and only if s is a moment sequence of a measure in J/*(IR). This establishes Hamburger 's characte- rization of moment sequences.

2. Case of One Polynomial

Suppose that a sequence s has the representation

s , = .(x"d~(x), n>O,

PolynomiaUy Positive Definite Sequences 489

for #e~'*(IR). Using (1) we find for p e a l that

(p(E) s), = ~ x"p(x) d#(x), n >= O. (3)

If p is supported by {p=0}, it follows from (3) that p(E)s is a positive definite sequence. This remark shows that condition (ii) of the introduction implies (i) for arbitrary Pl . . . . . Pt. In the case I= 1 we shall now formulate conditions on P=Pl assuring that (i) =~ (ii).

Theorem 1. Let p be a real polynomial and K = {p > 0}. Every sequence s for which s and p(E)s are positive definite has a representation

s.= ~x"d#(x), n>O,

with #e./g*(K), if and only if p has at least one of the following properties (a) K is compact, (b) deg(p) < 2, (c) K = ~

The proof of Theorem 1 will be prepared through a series of lemmas. We define

Cp= {a+ bpla, b e d + }, (4)

which is a convex cone contained in + d~p>= o~, and Cp is stable under multiplication.

Lemma 1. The dual cone of Cp is

• - {se ~ l s , p(E)se ~} . C p -

Proof. Let s e ~ , b e d . An easy calculation shows that

(s, bp) = (p(E)s, b) , (5)

so it follows that se Cp l if and only if

(s,a)>=O and (p(E)s,b)>O

for all a, b e d +, which is equivalent to s,p(E)se~. []

Lemma 2. Let p E ~ be such that K = {p => 0} is compact. I f s and p(E)s are positive definite there exists 12e Jg*(K) such that

s, = ~ x" du(x), n > 0 . K

Proof. By Hamburger's theorem there exist measures #, ve Jt'*(IR) such that

s,=~x"du(x), (p(e)s).=Ix"dv(x), n>O.

By (3) we have

(p(E) s), = I x"p(x) d#(x) = I x"p(x) d#(x) + I x"p(x) d#(x), K ~\K

so that

x"p(x) d,(x) = ~ x" dv(x)- S x"p(x) d,(x), K ~\K


which shows that the two positive measures

l r P d # and v - l e \ rPd # (6)

have the same moments . Since the first measure in (6) has compact support , the momen t problem is determinate, so the two measures in (6) are equal, hence v = p d#. The signed measure p d# being positive implies that supp(p)_~ {p>0} = K. [ ]

Lemma 3. Let pe sJ be such that K = {p>0} is non-compact. Then Cp oiven by (4) is closed in ~r

Proof. We shall use the same technique as in the proof of Theorem 3 of [3] and establish first that C j ~ r d is closed in d d for every d >0. Since K is non-compact , if the degree of p is even, then the coefficient of the highest degree term is positive. Thus for either pari ty of the degree, { p > l } is unbounded and deg(a+bp) =max(deg(a),deg(bp)) for a, b e s r +. In part icular q = a + b p e C f ~ J a implies a, b e d d. Let H be a subset of {p> 1} consisting of d + 1 different points and assume q , = a , + b , p , n = l , 2 . . . . , is a sequence from CpC~Sr d converging in d d to q~ ~r Then

0 < a.(h), b,(h) <= q.(h)~q(h)

for h e H , and it follows that (a,(h)) and (b,(h)) are bounded sequences for h~H. There exists a subsequence (n j) of (n) such that lim a,j(h) and l imb , (h) exist for

j ~ j ~ c ~ J h e l l .

If a sequence (r,) of polynomials from d d converges in d + 1 different points, the limit of (r,) exists in d d in the canonical topology of N d. It follows that there exist polynomials a, b e d d such that a , - ~ a and b,j-~b in d d, hence a, b e d § and q = a + bp, which shows that q~ CpC~d d. The assertion follows since a convex set C in ~ is closed if and only if C c ~ d is closed in ~d for all d ~ 0 (cf. [10, Example 7, p. 69]). [ ]

Lemma 4. Let p(x) = x 2 - 1 and K = IR\] - I, 1 [. Then Cp = d ~ .

Proof. Let q e ~ and let x ~ < X z < . . . < x . be the real roots of q of odd multiplicity. We shall prove that qe Cp.

We have x 1 . . . . . x ,e [ - 1, 1] a n d r is even. Fur thermore q admits a factorization q(x) = a(x) ( x - x 1)...(x - x~), where a e ~ +. Since d § ~ Cp and Cp is stable under multiplication, it suffices to prove that ( x - xx) ( x - x2)~ Cp for - 1 < x I < x 2 < 1. If we can find a constant k=k(x~,x2)~]O, 1] such that

( x - xl) ( x - x 2 ) - k(x 2 - 1)e ~r (7)

then the assertion will follow. In case x ~ + x 2 = O it suffices to define k=lx l l . I f x a +x2:#O and ke]0 , 1[, the

min imum of the polynomial in (7) is

re(k) = x~ x2 + k - (1 - k)- ~,

and we have to find ke ]0 , 1[ such that m(k)>O.

Polynomially Positive Definite Sequences 491

If �89 1 +x2)e]0 , 1[ then k 0 := 1 - �89 1 q- X2)e]0 , 1[ and

m(ko) = (1 - xl) (1 - x2) > O.

If �89 I +x2)e ] - 1,0[ then k 1 := 1 + �89 1 + x/)e]O, 1[ and

m(k~)=(l+xl)(l+x2)>O. []

Remark. If qes r satisfies q(x)>O for x r 1[, then q is of even degree say 2n, and following Svecov [12] q may be written q(x)= a(x)Z+ (x a - 1)b(x) 2 with ae d , , b e d , _ 1, which is stronger than the assertion in Lemma 4. Svecov's result may be deduced from a result of Luk~cs (cf. [-9, Problem 47, p. 78]) applied to the

reciprocal polynomial x2"q(xl-), which is non-negative for x e [ - 1 , 1]. This poly-

nomial is however only of degree 2n if q(0)4:0, so if q(0)=0 one has to apply Lukfics' result to q~(x)= q(x)+ e, e > 0, and finally let e--,0.

Lemma 5. Let peal be such that K= {p>0} is non-compact. Then C , = d ~ if and only if p satisfies either (b) or (c) of Theorem 1.

Proof. If K=IR then clearly C ,=~r ~ = d +.

Suppose that (b) holds. If p is a constant k, then necessarily k _-> 0 (the empty set is compact), so K =N. For p(x)=x then K = [0, oo[, and by [9, Problem 45, p. 78] every q e dt~" ~ot has the form q(x)= a(x)+ b(x)x with a, b e d +, i.e. Cp = d ~ . Using an affine transformation it is easy to see that Cp = d ~ for an arbitrary polynomial p of degree one. If p is of degree two and K ~IIL then p(x)=c(x-~)(x- f l ) with c > 0 and s < f l so K=lR\]s, fl[. Our assertion is that any real polynomial q satisfying q(x)> 0 for x e IRa]e, fl[ can be written q = a + bp with a, be d +. Using a suitable affine transformation, this is however a consequence of Lemma 4.

Suppose finally that neither (b) nor (c) are satisfied. Then p has degree n > 3 and has at least one real root of odd multiplicity. Let a, be the coefficient of the term x". If a, > 0 let fll denote the biggest root of odd multiplicity. Then p(x)>=0 for x >i l l and there exists flo<fll such that p(x)<0 for xe[flo, fll [. The polynomial q(x) = ( x - flo) ( x - ill) belongs to d ~ , but as we shall see, qr Cp. In fact if q = a + bp is possible with a, b e d +, then b = 0 since p has degree n > 3 with an>0, but this leads to the contradiction q = a e d +. If an<0 then n must be odd because K is assumed to be non-compact. Let s o denote the smallest zero of p of odd multiplicity. Then p(x)> 0 for x < s o and there exists el > So such that p(x)< 0 for x e ]s0, a 1]. The polynomial q(x)= ( x - % ) ( x - a 1) belongs to ~r but as we shall see, qr In fact i fq=a+bp is possible with a, b e d +, then since a,b are of even degree and p is of odd degree we conclude that 2 = deg(q)= max(deg(a), deg(bp)), hence b=0. This leads to the contradiction q = a e s t +. []

Proof of Theorem l. Suppose that p e d satisfies one of the conditions (a)-(c) and that s, p(E)se ~. If (a) is satisfied the integral representation of s follows from Lemma 2. If (a) is not satisfied and (b) or (c) are satisfied, the integral repre-

Z + .1_ sentation follows from Lemma 5 since se C o = ( d ; ) . If peal does not satisfy any of the three conditions (a)-(c) then Cv is a proper

subset of ~1~ by Lemma 5, and Cp is closed in d by Lemma 3. Let q e d ~ \ C v be


chosen. By Hahn-Banach's theorem there exists s~ ~ such that

< s , q ) < 0 and s~C~.

The first condition shows that s is not a moment sequence of a measure #e dC*(K), and the last condition and Lemma 1 show that s,p(E)se~. []

Remark. Note that Theorem 1 cannot be strengthened in the affirmative case to insist that all representing measures for s be supported by {p > 0}, since there exists a determinate Stieltjes moment sequence, which is not a determinate Hamburger sequence.

l + .1_ The first statement in Theorem 1 can be rephrased as C v =c(~r . Since Cp_C- ~r we always have • + • C v ___(~r , so the statement is equivalent with

.1_ + J . • J _ + J. Cp =(~r ) . On the other hand we have Cp =(Cv) l, so C v -(~1~ ) if and only if Cp = ~r by a Hahn-Banach argument as above. We have therefore proved:

Corollary 1. Let pea l and K={p=>0}. Then Cv=~r if and only if p satisfies at least one of the conditions (a)-(c) of Theorem 1.

Below we formulate special cases of Theorem 1, by indicating necessary and sufficient conditions for a sequence s to be moment sequence of a measure #~ ~'*(K), where K is one of the following subsets of IR:

K = ~ :

K = [ 0 , oo[:

K = [ a , oo[:

K = ] - oo,/~]: K=[~,/~]: K = [ - I , I ] : K = [ O , i ] : K = m ~ , fl[:

K=[~, f l ]w[~, f ] ,~<f l<7<6:

s ~ '

s, E s ~

s, E s - cts~ ~

s, f l s - Ese ~

s, ( g - aI) ( f l I - E)se

s,s--E2sE# s, E(I - E)s ~

s, ( E - QtI) ( E - fl I)se

s, - ( E - ~I) ( E - f i t ) (E- 71) (E- 6I)se ~ .

The cases K = I R and K = [ 0 , oo[ are the well-known results of respectively Hamburger and Stieltjes. The cases K = [ a , oo[ and K = ] - o o , fl] appeared in Devinatz [5]. Finally the cases K = [ - 1 , 1 ] and K=IR\]a , fl[ appeared in respectively Haviland [7] and Svecov [12]. For another discussion of the case K = [ - 1, 1] see also Atzrnon [2].

For a polynomial p not satisfying any of the three conditions of Theorem 1 there exists a sequence se ~ violating the first statement of the Theorem. We will offer an example which does not depend upon the Hahn-Banach theorem.

Example. Let p(x)=x 3 so K = [0, oo[. We will find s e ~ such that E3se~, but the representing measure of s, which is uniquely determined in Jg*(F,), is not concentrated on [0, oo [.

Let t be an indeterminate Stieltjes moment sequence and let # be a Nevanlinna extremal measure representing t such that supp(#)c~]- 0% 0[~= 0 and 0~supp(#).


The existence of such a measure follows from Nevanlinna's parametrization (cf. [1] or [11]). The measure a = x - 2 d p ( x ) is determinate (cf. [13]) and is not concentrated on [0, ~ [ . The sequence s of moments of a has the desired property since E2s = t is a Stieltjes moment sequence and therefore Et = E 3 s ~ ~ . Note that C~ is closed and xe~r oot\C~.

3. Case of Several Polynomials

As remarked in Sect. 2, condition (ii) of the introduction implies condition (i). We shall now state a sufficient condition on PI ... . . Pz which assures the converse implication.

Theorem 2. Le t P l, ..., Pl be real polynomials and suppose that K = {Pl >O}~...c~{pl>-O } is compact.

Every sequence s such that s, p l (E)s , . . . ,p l (E)se:~ is a moment sequence o f a measure # e J /* (K) .

Proof. Let s e ~ be such that p j ( E ) s ~ for j = 1 .. . . . I.

Suppose first that at least one of the sets {pj>_-O} are compact, say that {Pl >0} is compact. By Theorem 1 we then have

sn = S x"d#(x ) , n > 0

for #eJg*({pl >0}). By (3) we have

(pj(E) s)n = ~ x~pj(x) dp(x) , n > O,j = 2 . . . . . l,

and by Hamburger 's theorem there exist measures vf i JI*(IR) such that

(pj(E) s), = ~ xndv~(x), n > O,j = 2 . . . . . I.

This implies that the two measures in sg*(IR)

l~p~ >= ojPfl# and v j - l~p~ < o~pjd#

have the same moments, and are therefore equal since the first measure has compact support contained in supp(g)=C{p~>0}. It follows that p f l p = v j for j = 2 .... . l, and since vi is a positive measure, we have supp(#)C_{p~>0} for j = 2,..., l, and the assertion follows.

Suppose next that all the sets {pj > 0} are non-compact. If each pj were of even degree, then the leading coefficients would be positive, so that each set {p j>0} would contain a set of the form ~ , \ [ - ct, a]. Since K is assumed to be compact, we see that at least two of the polynomials must be of odd degree with highest term coefficients of different sign. Without loss of generality we may assume that

P l ( X ) = xEn+ l --b aEnXEn-I - ... + a 0

and

p2(x) = - x 2m+ ~ + b2mx 2m + . . . + bo.


For t E ~ then E2kte 9 a for k >=0. We may therefore change the polynomials Pl, P2 to two polynomials ql,q2 of the same odd degree, say 2n+ 1, such that q~(E)s, q2(E)s~9 a and such that

ql(x)=x2"+l+c2,x2"+. . .+c o , q2(x) = - x 2 " + l + d 2 . x 2 " + . . . + d o

with c2,, d2,>0. We define q=q2q2+qlq2=q~q2(q ~ +q2), which is a polynomial of even degree 6n + 2. The coefficient of the highest degree term is - ( c2 , + dz,), which is negative, so A = { q > 0 } is compact. Furthermore q(E)sE~. In fact for a6a r + we have by (5) that

(q(E)s, a) = (s, qa) = (ql(E)s, q22a ) + (q2(E)s, q2a) >= O.

By Lemma 2 there exists a measure #~ J/*(A), which has compact support, such that

s.= ~. x"d~(x), n >=0.

An argument similar to the one in the first part of the proof shows that supp(p)__c {pj>O} for all j = 1 . . . . ,t, hence supp(p)_---K. []

Remark. With the notation of Theorem 2 we can introduce the cone

Cp ...... p, = {a o + atp 1 + . . . +alptlaj~ al+,j = O, ..., l},

which is a subset of ag~. It is easy to see that the dual cone of Cp ...... p, is

C • = {s~ galpj(E)se~,j = 1 ..... l} P l , . . . , P l

By an argument similar to the one leading to Corollary 1 we obtain :

Corollary 2. Suppose K = {Pl >0}c~...c~{p I >0} is compact. Then

, . , . , P l - - K "

Corollary 3. Let ~ < ft. A necessary and sufficient condition for a sequence s to be moment sequence of a measure #EJ/r is that (E-c~I)s and ( f l I -E)s are positive definite.

Proof. In fact, defining p l ( x ) = x - ~ , p z ( x ) = f l - x we have {pl>O}n{p2>O} = [cq/~]. If (E-~I)s , ( f l l - E ) s E ~ We get by addition that ( f l - c 0 s ~ and hence sE ~a, so the assertion follows from Theorem 2. []

Corollary 3 is in Devinatz [5]. For the case of [0, 1] see also Atzmon [2], and for a generalization to semigroups see Berg and Maserick [4].

As an application of Theorem 2 and Corollary 3 we can state" Let ~ </~ < 7 < 6. Then s is a moment sequence for a measure I~Jll*([~,fl]wEy,6]) if and only if ( E - el)s, ( 6 I - E)s and ( E - ill) ( E - 71)s are positive definite.

4. The Multidimensional Case

A generalization of the above difficulties (see however [8]).

results to several dimensions presents some


Let s :]N~ ~ I R be a m u l t i s e q u e n c e a n d let p = p(x 1 . . . . . x k) be a real p o l y n o m i a l in k var iab les , k > 2. I f E j, j = 1 . . . . , k deno t e s the un i t shift in the f t h coo rd ina t e ,

p(E) =p(E~ , . . . , E k) is a wel l -def ined shift ope ra to r . T h e first ser ious p r o b l e m is t ha t H a m b u r g e r ' s t h e o r e m does n o t genera l ize to k d imens ions , i.e. there exist n o n - nega t ive p o l y n o m i a l s wh ich are n o t sums of squares , a n d there exist pos i t ive

def ini te mu l t i s equences wh ich are n o t m o m e n t m u l t i s e q u e n c e s (cf. [3]). I f we

h o w e v e r rep lace the pos i t ive def ini teness c o n c e p t by the s t ronge r c o n c e p t o f be ing

a m o m e n t mul t i s equence , then L e m m a 2 a n d its p r o o f can be ca r r i ed o v e r to yield

the fo l lowing :

T h e o r e m 3 . Le t p be a real polynomial o f k variables such that K = { p > 0 } is

compact. I f s and p(E)s are moment multisequences, then there exists a positive measure # o f compact support contained in K such that

�9 ., ,~ ,k ..., nk> 0 s(nl, . nk)= ~ X I . . . x k d#(x) , nl ,

T h e asser t ion in T h e o r e m 3 migh t still h o l d u n d e r the w e a k e r a s s u m p t i o n o f s a n d p(E)s be ing pos i t ive def ini te and K = { p = 0 } be ing compac t . Th is e.g. t rue if

2 a n d a j > 0 for j = l , . . . , k p(x) = 1 - a l x ~ - . . . - akX k

(cf. [8]). F o r o t h e r e x a m p l e s in this d i r e c t i o n see [8].

References

1. Akhiezer, N.I. : The classical moment problem. Edinburgh : Oliver and Boyd 1965 2. Atzmon, A. : A moment problem for positive measures on the unit disc. Pac. J. Math. 59, 317-325

(1975) 3. Berg, C., Christensen, J.P.R., Jensen, C.U. : A remark on the multidimensional moment problem.

Math. Ann. 243, 163-169 (1979) 4. Berg, C., Maserick, P.H.: Exponentially bounded positive definite functions. To appear in Ill. J.

Math. 5. Devinatz, A.: Integral representations of positive definite functions. Trans. A.M.S. 74, 56-77 (1953) 6. Haviland, E.K. : On the momentum problem for distributions in more than one dimension. Am. J.

Math. 57, 562-568 (1935) 7. Haviland, E.K. : On the momentum problem for distribution functions in more than one

dimension. II. Am. J. Math. 58, 164-168 (1936) 8. McGregor, J.L. : Solvability criteria for certain N-dimensional moment problems. J. Approxima-

tion Theory 30, 315-333 (1980) 9. Polya, G., Szeg6, G.: Problems and theorems in analysis, Vol. II. Die Grundlehren der mathemati-

schen Wissenschaften, Bd. 216. Berlin, Heidelberg, New York: Springer 1976 10. Schaefer, H.H.: Topological vector spaces. Graduate Texts in Mathematics, Vol. 3. Berlin,

Heidelberg, New York: Springer 1971 11. Shohat, J.A., Tamarkin, J.D. : The problem of moments. Mathematical Surveys, Vol. 1. New York :

American Mathematical Society 1943 12. Svecov, K.I.: On Hamburger's moment problem with the supplementary requirement that masses

are absent on a given interval. Commun. Soc. Math. Kharkov 16, 121-128 (1939) (in russian) 13. Wright• F.M. : •n the backward extensi•n •f p•sitive definite Hamburger m•ment sequences. Pr•c.

A.M.S. 7, 413-422 (1956)

Received October 12, 1981

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Polynomially positive definite sequences