Discrete analytic functions, structured matrices and a new family of moment problems

TL;DR

This paper explores the connections between discrete analytic functions, structured matrices, and moment problems, utilizing Krein space realizations and Moebius transforms to analyze functions in the upper right quadrant.

Contribution

It introduces a novel approach linking discrete analytic functions with structured matrices and moment problems through Krein space realizations and function theory in the unit disk.

Findings

Discrete analytic functions can be associated with positive measures on [0,2π].

Rational cases are characterized by rational Carathéodory and spectral functions.

Unitary dilations in Krein spaces connect rational and general cases.

Abstract

Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An important role is played by a Krein space realization result of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the origin. A key observation is that, using a simple Moebius transform, one can reduce the study of discrete analytic functions in the upper right quadrant to problems of function theory in the open unit disk. As an example, we associate to each finite positive measure on $[0,2\pi]$ a discrete analytic function on the right-upper quarter plane with values on the lattice defining a positive definite function. Emphasis is put on the rational case, both when an underlying Carath\'eodory function is rational and when, in the positive case, the spectral function is rational. The rational case and the general case are linked via the existence of a unitary dilation, possibly in a Krein space