Multidimensional vector-valued $Z$-transform and its applications

TL;DR

This paper develops a comprehensive theory of the multidimensional vector-valued $Z$-transform, exploring its structure, properties, and applications to abstract Volterra difference equations and convolution products in locally convex spaces.

Contribution

It introduces a systematic study of the multidimensional vector-valued $Z$-transform, extending classical results to functions in locally convex spaces with applications to difference equations.

Findings

Structural characterizations of the multidimensional vector-valued $Z$-transform

Applications to abstract Volterra difference equations

Analysis of multidimensional discrete convolution in vector-valued setting

Abstract

In this paper, we systematically investigate the multidimensional $Z$-transform of functions with values in sequentially complete locally convex spaces over the field of complex numbers. We provide many structural characterizations, remarks and applications of established results to abstract Volterra difference equations depending on several variables. We also consider multidimensional discrete convolution products in vector-valued setting.