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.

Monotonicity of zeros of orthogonal Laurent polynomials

“…For any given strong distribution in (a, b) there exists a unique sequence, up to a nonzero constant factor normalization, of polynomials {B n } ∞ 0 such that B n is a polynomial of precise degree n and B n satisfies the relations The first systematic study of these polynomials, which may be called orthogonal Laurent polynomials or simply orthogonal L-polynomials was done by Jones, Thron and Waadeland [10] in connection with the so-called strong Stieltjes moment problem.…”

Unique solvability of the strong Hamburger moment problem

“…There will not be a unique solution to the SSMP if the positive T-fraction terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870002646X 8 Olav Njastad and W. J. Thron [4] determined by the moments diverges (see [9]). Hence for this sequence of moments the SHMP also must have an infinite number of solutions.…”

“…PROPOSITION terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870002646X [9 ] Unique solvability of the strong Hamburger moment problem 13 ROOF. Note that Q n (z) and P n (z) are L-polynomials with real coefficients.…”

A matrix approach to the computation of quadrature formulas on the unit circle

“…Here we will also propose as an estimation for I µ (f ) an n-point quadrature rule I n (f ) = n j=1 λ j f (z j ) with distinct nodes on the unit circle but now imposing exactness not for algebraic polynomials but trigonometric polynomials or more generally Laurent polynomials or functions of the form L(z) = q j=p α j z j , α j ∈ C, p and q integers with p ≤ q. Now, it should be recalled that Laurent polynomials on the real line were used by Jones and Thron in the early 1980 in connection with continued fractions and strong moment problems (see [27] and [29]) and also implicitly in [30]. Their study, not only suffered a rapid development in the last decades giving rise to a theory of orthogonal Laurent polynomials on the real line (see e.g.…”