Document Type

Article

Publication Date

7-18-2022

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, 2π] 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 Carathéodory 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.

Comments

NOTICE: this is the author’s version of a work that was accepted for publication in Bulletin des Sciences Mathématiques. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Bulletin des Sciences Mathématiques, volume 179, in 2022. https://doi.org/10.1016/j.bulsci.2022.103175

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Elsevier

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This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

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