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A general quantified Ingham-Karamata Tauberian theorem

Gregory Debruyne

Abstract

We provide a general quantified Ingham-Karamata Tauberian theorem with a flexible one-sided Tauberian condition under several types of boundary behavior for the Laplace transform. Our results in particular improve a theorem by Stahn, removing a vexing restriction on the growth of the Laplace transform. Improving existing optimality results, we also show that the obtained quantified rate is optimal in almost all cases.

A general quantified Ingham-Karamata Tauberian theorem

Abstract

We provide a general quantified Ingham-Karamata Tauberian theorem with a flexible one-sided Tauberian condition under several types of boundary behavior for the Laplace transform. Our results in particular improve a theorem by Stahn, removing a vexing restriction on the growth of the Laplace transform. Improving existing optimality results, we also show that the obtained quantified rate is optimal in almost all cases.

Paper Structure

This paper contains 28 sections, 11 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Statement of the main theorem
  3. Preliminaries
  4. The Tauberian condition
  5. The hypotheses on the Laplace transform
  6. The main theorem
  7. The Tauberian argument: Estimation of the high frequencies
  8. The key lemma
  9. Existence of test functions
  10. Application: the unquantified Ingham-Karamata theorem
  11. Application: the Berry-Esseen inequality
  12. Tauberian conditions with higher-order derivatives
  13. The estimation of the low frequencies: Completion of the proof of Theorem \ref{['thrikimain']}
  14. Translation of the convolution: the exact behavior
  15. Differentiable functions: the class $\mathcal{T}_{\mathrm{Dif}}$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $S: [0,\infty) \rightarrow \mathbb{C}$ be a Lipschitz continuous function. If the Laplace transform $\mathcal{L}\{S;s\} := \int^\infty_{0} e^{-su} S(u) \dif u$, initially convergent on the half-plane $\operatorname{Re} \: s > 0$, admits a continuous extension to the line $\operatorname{Re} \: s

Theorems & Definitions (22)

  • Theorem 1.1: Ingham, Karamata
  • Theorem 1.2: Stahn
  • Theorem 1.3
  • Theorem 1.4: Quantified Wiener-Ikehara with simple pole
  • proof : Deduction from Theorem \ref{['coran']}
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 12 more