On the solution of moment problems by reproducing kernel methods (Q1098047)

The author presents a general method for finding necessary and sufficient conditions in order that a complex-valued function be represented as an integral of a given integrand with respect to some nonnegative measure on the real axis. Let s range over a subset of the complex numbers \({\mathbb{C}}\), \(\lambda\) range over the real numbers \({\mathbb{R}}\), and h(s,\(\lambda)\) be a given complex-valued function. Starting with a given integrand h(s,\(\lambda)\), the author proceeds by constructing a kernel \(k(s_ 1,s_ 2)\) from h(s,\(\lambda)\) considered as a function of its first argument. Only certain operations are allowed in this construction which are described in detail. This problem of constructing the correct kernel for h is at the heart of this approach and is fundamentally linked to the problem of solving a certain class of functional equations. The author presents this class along with ten solutions for various parameter choices. Mimicking the kernel construction for a candidate function f(s) leads to the main condition for representability, namely that the resulting kernel be positive (definite or indefinite). The proof draws on the theory of reproducing kernel spaces and the paper contains the appropriate information on these spaces. Overall the author's approach is a continuation and an extension of that taken by A. Devinatz in a series of papers in the period 1950-1960. However, the main constructions in those works can be done once and for all by using the Nagy Principle Theorem [cf. \textit{A. Devinatz}, J. London Math. Soc. 35, 417-424 (1960; Zbl 0097.109)]. The author presents a new generalization of the Nagy Principal Theorem appropriate to the extensions developed in this interesting paper. Applications to the solution of the moment problem \(f(s)=\int^{\infty}_{- \infty}h(s,\lambda)dw(\lambda)\) are given.