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Uniformity in the Fourier inversion formula with applications to Laplace transforms

Joannis Alexopoulos

TL;DR

This work addresses when local uniform convergence holds for Fourier and Laplace inversion in Banach-space-valued settings. By developing a dimension-aware Fourier inversion framework using the Dirichlet kernel and spherical means, it proves local uniform convergence under regularity and compactness conditions, distinguishing odd and even dimensions. The authors then translate these Fourier results to Laplace inversion, obtaining local uniform convergence for $F$-valued functions and extending classical semigroup inversion results to Favard spaces. The findings enlarge the toolkit for analyzing linear evolution equations and contribute to a sharper understanding of when inversion formulas preserve uniformity in $t$, with direct implications for $C_0$-semigroups and their Favard-space regularity.

Abstract

We systematically find conditions which yield locally uniform convergence in the Fourier inversion formula in one and higher dimensions. We apply the gained knowledge to the complex inversion formula of the Laplace transform to extend known results for Banach space-valued functions and, specifically, for C_0-semigroups.

Uniformity in the Fourier inversion formula with applications to Laplace transforms

TL;DR

This work addresses when local uniform convergence holds for Fourier and Laplace inversion in Banach-space-valued settings. By developing a dimension-aware Fourier inversion framework using the Dirichlet kernel and spherical means, it proves local uniform convergence under regularity and compactness conditions, distinguishing odd and even dimensions. The authors then translate these Fourier results to Laplace inversion, obtaining local uniform convergence for -valued functions and extending classical semigroup inversion results to Favard spaces. The findings enlarge the toolkit for analyzing linear evolution equations and contribute to a sharper understanding of when inversion formulas preserve uniformity in , with direct implications for -semigroups and their Favard-space regularity.

Abstract

We systematically find conditions which yield locally uniform convergence in the Fourier inversion formula in one and higher dimensions. We apply the gained knowledge to the complex inversion formula of the Laplace transform to extend known results for Banach space-valued functions and, specifically, for C_0-semigroups.

Paper Structure

This paper contains 10 sections, 15 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Uniformity in the Fourier inversion formula
  3. Preliminaries
  4. Compactness criteria and result on Lg
  5. Uniformity in Lg with odd Lg
  6. Uniformity in Lg with even Lg
  7. Applications to Laplace transforms
  8. From Fourier to Laplace inversion
  9. Application to Lg
  10. Function spaces

Key Result

Proposition 1.1

If $F \in H_{\textrm{loc}}(\mathbb R_+;X)$For the definitions of function spaces see Appendix appendix, with $F(0) = 0$, has exponential growth bound $\omega_0 \in \mathbb R$, then for any $\omega>\omega_0$.

Theorems & Definitions (31)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Remark 1.4
  • Proposition 2.1: pinsky_book
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: A uniform version of the Riemann-Lebesgue lemma
  • ...and 21 more