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Solutions of the Strong Hamburger Moment Problem

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Solutions of the Strong Hamburger Moment Problem JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 197, 227]248Ž. 1996 ARTICLE NO. 0017 Solutions of the Strong Hamburger Moment Problem Olav Njastad*Ê Department of Mathematical Sciences, Uni¨ersity of Trondheim}NTH, N-7034 Trondheim, Norway Submitted by George Gasper View metadata, citation and similar papers at core.ac.uk brought to you by CORE Received December 27, 1993 provided by Elsevier - Publisher Connector The strong Hamburger moment problem for a bi-infinite sequence Äcn: n s 0, " 1, " 2, . .4Ž. can be described as follows: 1 Find conditions for the existence of a Ž.m Ž.` ` H`nmŽ. Ž. positive measure on y , such that cn s y` td tfor all n. 2 When there is a solution, find conditions for uniqueness of the solution.Ž. 3 When there is more than one solution, describe the family of all solutions. In this paper a theory concerning questionŽ. 3 is developed. In particular, an analog to the Nevanlinna parametrization describing the solutions of the classical Hamburger moment problem is given. Q 1996 Academic Press, Inc. 1. INTRODUCTION The classical Hamburger moment problem can be defined as follows: A Ä4 Ž. sequence cn: n s 0, 1, 2, . of real numbers is given. 1 Find conditions Ž. Ž. for there to exist a positive measure m on y`, ` such that cn s H` n mŽ. Ž. y` td tfor n s 0, 1, 2, . 2 When there is a solution, i.e., when at least one such measure m exists, find conditions for uniqueness of the solution.Ž. 3 When there is more than one solution, describe the family of all solutions. The problem is called determinate when there exists exactly one solution, indeterminate when there exists more than one solution. The problem was first discussed by Stieltjeswx 28 for the case that m has support inw 0, `.Žthe classical Stieltjes moment problem . , and then by Hamburger wx9 for the general case that the support of m is only required to be contained in Ž.y`, ` . This initial work was followed by an extensive development of a theory of moment problems, where the connection with the theory of orthogonal polynomials plays a central role. We refer the reader to the papers and monographswx 1, 6, 9, 13, 17, 19, 20, 26]30 . U Research supported by the Norwegian Research Council. 227 0022-247Xr96 $12.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 228 OLAV NJASTADÊ The strong Hamburger moment problemŽ and strong Stieltjes moment problem. can be formulated in the same way as the classical problem, Ä4 except that here bi-infinite sequences cn: n s 0, " 1, " 2, . are in- volved. This problem was introduced by Jones and Thron about 1980 wx14]16Ž and for the Stieltjes case by Jones et al.wx 17. A theory of these problems and their connection with orthogonal Laurent polynomials was developed in the ensuing years as far as questionsŽ. 1 and Ž. 2 are con- cerned. Seewxwx 7, 10]12, 23]25 . Also in 2 the strong moment problem was briefly discussed. In this paper we continue these investigations to develop a theory concerning questionŽ. 3 , and present an analog of the Nevanlinna parametrization in the classical case. The classical and strong moment problems are special cases of a more general theory, where moments corresponding to an arbitrary countable sequence Ä4an of points are involvedŽ in the classical and strong cases the points are Ä4` repeated and Ä`, 0 4 cyclically repeated, respectively. , and where orthogonal rational functions play the role of orthogonal polynomi- als and Laurent polynomials. For an introduction to orthogonal rational functions and discussions of questionŽ.Ž 1 and question Ž. 2 in the case of a finite number of points cyclically repeated. , seewx 3]5, 8, 21, 22 . 2. PRELIMINARIES The basic theory of orthogonal Laurent polynomials and strong moment problems including the results sketched in this section can be found in wx7, 10]12, 14]17, 23]25 . Ž. For any pair p, q of integers with p F q, let H p, q denote the complex j linear space spanned by the functions z , j s p,...,q. We shall write H 2 m s Hym, m and H 2 mq1 s HyŽmq1., m for m s 0, 1, 2, . , and H s ` D ns0 H n. An element of H is called a Laurent polynomial. Let M be a linear functional on H, and assume that M is positive on Ž.Žy`,` i.e., MLwx)0 for all L g H where LzŽ.k0, Lz Ž.G0 when Ž.. n zgy`,`. The moments cnnare defined for all n g Z by c s Mzwx. The strong Hamburger moment problem Ž.SHMP consists of finding all m Ž.` ` H` Ž.u mu Ž. positive Borel measures on y , such that MLwxs y` L d n H` un muŽ. for all L g H, or equivalently such that Mzw xs y` d for all n g Z. A necessary and sufficient condition for there to exist at least one such measure is that M is positive on Ž.Žy`, ` as defined above. The problem is called determinate if there is only one solution, indeterminate otherwise. ²: An inner product , is defined on HRR= H by ²:P,QsMPzŽ.?Qz Ž..2.1 Ž . STRONG HAMBURGER MOMENT PROBLEM 229 Ž j Here HR denotes the real space spanned by z , j s 0, " 1, " 2, . By orthonormalization of the baseÄ 1, zy1, z, zy22, z ,..., zynn, z ,...4 , or- thonormal Laurent polynomials Ä4wn are obtained. They have the form q 2 m , ym m w2mŽ.z sq??? qqz2m,m ,q2m,m)02.2 Ž . zm q 2mq1,yŽmq1. m w2m 12Ž.z sq??? qqzm1, m , q zmq1 q q2mq1,yŽmq1.)02.3Ž. for m s 0, 1, 2, . We shall in the following assume that M gi¨es rise to a regular system, which means that q2 m, ym / 0, q2 mq1, m / 0 for all m. The associated orthogonal Laurent polynomials Ä4cn are defined by wunnŽ.yw Ž.z cnŽ.zsM .2.4 Ž . uyz Ž.The functional is applied to its argument as a function of u. We note that c c c 02' 0, mg Hym,my12, mq1g HyŽmq1.,my1. We further define quasi-orthogonal Laurent polynomials wnŽ.z, t and their associated Ž. Ž . Laurent polynomials cnz, t of order nns2mand n s 2m q 1by w t w tw 2mŽz, .s 2m Ž.zy z2my1 Ž.z Ž2.5 . t w2m 12Žz,t .swm12 Ž.zy wm Ž.z Ž2.6 . q qz ctctc 2mŽz, .s2m Ž.zyz2my1 Ž.z Ž2.7 . t c2m 12Žz,t .scm12 Ž.zy cm Ž.z Ž2.8 . q qz Here t g RÃs R j Ä4` . For n s 1, 2, . we may also write wunnŽ.,tyw Ž.z,t cnŽ.z,tsM .2.9 Ž. uyz The quasi-approximants RnŽ. z, t are defined by cnŽ.z, t RznŽ.,ts . Ž. 2.10 wnŽ.z,t For t s 0 they are simply called approximants. Ž. All zeros of wn z, t are real and simple, and for t g R there are n of Ž.Žn.Ž. Žn.Ž. them while for t s ` there are n y 1 . Let zt1 ,...,ztn denote 230 OLAV NJASTADÊ these zeros. Then the following quadrature formulas are valid, where Žn. Žn. lt1 Ž.,...,ltn Ž.are positive weights: n Žn.Žn. MFwxsÝltkkŽ.FŽ.zt Ž. Ž2.11 . ks1 valid for F g Hy2 m,2my2 when n s 2m, for F g Hy2m,2m when n s 2mq1. In particular, 2m p p Ž2 m.Ž2m. MzwxsÝltztkkŽ. Ž. for p sy2m,...,2my2 ks1 Ž.2.12 2m1 q p p Ž2mq1.Ž2mq1. Mzwxs ÝltztkkŽ. Ž. for p sy2m,...,2m. ks1 Ž.2.13 Ž.u Žwu Žt .w Žt ..Žu . The function f s nn, y z, r yzbelongs to Hy2m,2my2 for n s 2m, and to Hy2 m,2m for n s 2m q 1. Therefore by the quadra- ŽŽŽn.Ž. ture formulas taking into account that wznk t,ts0 we may write n Žn. ltk Ž. RznŽ.,tsy Ý . Ž. 2.14 ztŽn.Ž. z ks1 k y It follows fromŽ.Ž. 2.12 ] 2.14 that 2my1 t k 2 m Rz2mŽ.,sy ÝczyŽkq1.qOz Ž.at 0, Ž. 2.15 ks0 2my1 1 yk Rz2mkŽ.,tsÝcz1qO at `, Ž. 2.16 yž/z2m ks1 2my1 t k 2m Rz2mq1Ž.,sy ÝczyŽkq1.qOz Ž.at 0, Ž. 2.17 ks0 2mq1 1 yk Rz2m1Ž.,tsÝczk1 qO at `. Ž. 2.18 q y ž/z2mq2 ks1 The following determinant formulas are valid: q2 m , m w c w c y z 2mŽ.z 2my12 Ž.z y z my12 Ž.z m Ž.z s Ž2.19 . qmy1,ym q2m 1, m w c w c q z 2mŽ.z 2mq12 Ž.z yz mq12 Ž.z m Ž.z s . Ž 2.20 . q2m,m STRONG HAMBURGER MOMENT PROBLEM 231 The following general Christoffel]Darboux formulas are valid for arbitrary complex coefficients a, b, c, d, and for z, z g C y Ä40: c w ? cz wz za 2my12Ž.z qb my12 Ž.z c m Ž .qd2m Ž . zczwz?c w yc2my12Ž.qdmy12 Ž.am Ž.zqb2m Ž.z 2my1 q2m,ym sŽ.Ž.ac y bd q z y z ÝŽacj Ž.z q 2my1,ym js0 qbwjjŽ.zc.Žcz Ž.qdwz j Ž. Ž2.21 . c w ? cz wz za 2mq12Ž.z qb mq12 Ž.z c m Ž .qd2m Ž .
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