A Strong Stieltjes Moment Problem

Abstract

Abstract.This paper is concerned with double sequences of complex numbers C = [cn}^x and with formal Laurent series Lq(C) = 2f -c_mzm and /."(C) = 2o°cmz~m generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having L0(C) and LX(C) as asymptotic expansions at z -0 and z = oo, respectively? (2) Does there exist a real-valued bounded, monotonically increasing function <f>(f) with infinitely many points of increase on [0, oo) such that, for every integer n, c" = / 5°(-0" dty(t)l The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function ¡f/(t) has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive 7"-fractions) which correspond to L0(C) at z = 0 and LX(C) at z = oo. Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have Lq(C) and LX(C) as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions ^°\t) and ^'"'(O solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive T-fraction that corresponds to both Lq(C) and ¿"(C).References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.

Magnetic Properties of Multiply Mn δ-Doped GaAs

“…The zeros of these polynomials behave in a way which is similar to the behaviour of zeros of polynomials which are orthogonal on the positive real line. The following lemma can be found in [9][10][11]16], but we include a proof for completeness.…”

“…This representation is also the key to the inverse spectral transform. It turns out that the function z f (z) can be written as a special continued fraction, known as a T -fraction, see [6,11].…”

“…Under the assumption that a n > 0 and b n > 0, it is known that the zeros of these polynomials are simple, real and positive; see [9][10][11]16] and lemma 2.1 below. The first set of spectral data we choose are the ordered zeros of the polynomial P N (z) = det(z M − L) which we denote by 0 < λ 1 < · · · < λ N .…”

“…Using T -fractions [6,11], we explain in section 2.2 how to obtain a n and b n when the spectral data λ j and w j are given. We also note that the polynomials P n satisfy…”

Solutions of the Strong Hamburger Moment Problem

“…14᎐16 and for the Stieltjes case by Jones et al 17 A theory of these problems and their connection with orthogonal Laurent polynomials was Ž . Ž .…”