Uniqueness sequences for multidimensional vector-valued Laplace transform

TL;DR

This work develops a theory of uniqueness sequences for the multidimensional vector-valued Laplace transform, linking it to the classical one-dimensional theory and to totality of exponential families in $L^1([0,\infty)^n)$. It rigorously defines the multidimensional transform for functions $f:[0,\infty)^n\to X$, with regions of convergence $\Omega(f)$ and $\Omega_{abs}(f)$ and introduces generalized convolutions, laying a framework to analyze uniqueness. Core contributions include establishing equivalences between multidimensional and one-dimensional uniqueness sequences, exploring subsequence and Müntz-type constructions, and formulating a multivariate Müntz-based partial solution to constructing multidimensional uniqueness sequences. A subordination principle shows how fractional power coordinate transforms yield new uniqueness sequences, with corollaries ensuring preservation of uniqueness under $\gamma$-powers and providing criteria relevant to inversion and applications to Volterra-type problems.

Abstract

In this research article, we consider the uniqueness sequences for multidimensional vector-valued Laplace transform. We establish the fundamental relationships between uniqueness sequences for one-dimensional Laplace transform and uniqueness sequences for multidimensional Laplace transform. We also provide several illustrative examples, open problems and useful observations in the above direction.

Theorems & Definitions (23)

• Theorem 1.1
• Definition 1.2
• Lemma 1.3
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5