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

Marko Kostic

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.

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

Abstract

In this paper, we systematically investigate the multidimensional -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.
Paper Structure (9 sections, 14 theorems, 92 equations)

This paper contains 9 sections, 14 theorems, 92 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Multidimensional $Z$-transform of vector-valued functions
  3. Discrete convolution products and multidimensional $Z$-transform
  4. The discrete convolution product $\ast_{{\rm D}}^{l,j}$
  5. Applications of multidimensional vector-valued $Z$-transform to abstract Volterra difference equations with multiple variables
  6. Abstract linear difference equations depending on several variables
  7. Abstract multi-term Volterra difference equations on ${\mathbb Z}^{n}$
  8. The abstract fractional difference equations with generalized Weyl $(a,m)$-fractional derivatives
  9. Conclusions and final remarks

Key Result

Proposition 2.3

Suppose that $\alpha,\ \beta \in {\mathbb C}$, $f: D \rightarrow X$, $g: D \rightarrow X$ and for each seminorm $p\in \circledast$ there exists a real constant $M_{p}>0$ such that the estimate est12 holds for any $z=(z_{1},z_{2},\ldots ,z_{n})\in \Omega ,$ and both sequences $f(\cdot),$$g(\cdot)$. T

Theorems & Definitions (29)

  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • proof
  • Definition 3.1
  • ...and 19 more