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Generalizations of Koga's version of the Wiener-Ikehara theorem

Bin Chen, Jasson Vindas

TL;DR

The paper generalizes the Wiener-Ikehara Tauberian theorem by imposing boundary conditions on the real part of the Laplace transform rather than full analytic continuation, extending Koga's 2021 results and replacing monotonicity with log-linearly slowly decreasing behavior. It develops a local pseudofunction boundary framework, proves Laplace-transform, Laplace-Stieltjes-transform, and power-series versions, and provides a rapid renewal-theorem proof (Blackwell) as a key application. Central contributions include new theorems ensuring $S(x)\sim a e^{x}$ under weaker hypotheses, and corollaries that translate these results to renewal theory and Dirichlet-series contexts. The approach broadens Tauberian tools, offering streamlined proofs and potential extensions to Beurling primes and related harmonic-analytic settings with practical impact on renewal theory and analytic number theory.

Abstract

We establish new versions of the Wiener-Ikehara theorem where only boundary assumptions on the real part of the Laplace transform are imposed. Our results generalize and improve a recent theorem of T. Koga [J. Fourier Anal. Appl. 27 (2021), Article No. 18]. As an application, we give a quick Tauberian proof of Blackwell's renewal theorem.

Generalizations of Koga's version of the Wiener-Ikehara theorem

TL;DR

The paper generalizes the Wiener-Ikehara Tauberian theorem by imposing boundary conditions on the real part of the Laplace transform rather than full analytic continuation, extending Koga's 2021 results and replacing monotonicity with log-linearly slowly decreasing behavior. It develops a local pseudofunction boundary framework, proves Laplace-transform, Laplace-Stieltjes-transform, and power-series versions, and provides a rapid renewal-theorem proof (Blackwell) as a key application. Central contributions include new theorems ensuring under weaker hypotheses, and corollaries that translate these results to renewal theory and Dirichlet-series contexts. The approach broadens Tauberian tools, offering streamlined proofs and potential extensions to Beurling primes and related harmonic-analytic settings with practical impact on renewal theory and analytic number theory.

Abstract

We establish new versions of the Wiener-Ikehara theorem where only boundary assumptions on the real part of the Laplace transform are imposed. Our results generalize and improve a recent theorem of T. Koga [J. Fourier Anal. Appl. 27 (2021), Article No. 18]. As an application, we give a quick Tauberian proof of Blackwell's renewal theorem.
Paper Structure (7 sections, 10 theorems, 49 equations)

This paper contains 7 sections, 10 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Application: Renewal theorems
  3. Exact Wiener-Ikehara theorem revisited
  4. Proof of Theorem \ref{['KWIthmain']} (and Theorem \ref{['KWIthStieltjes']})
  5. The power series case: proof of Theorem \ref{['KWIthpowerseries']}
  6. Proof of Corollary \ref{['KWIthIK']}
  7. Concluding remarks

Key Result

Theorem 1.1

Let $S\in L^{1}_{loc}[0,\infty)$ be log-linearly slowly decreasing and satisfy Let $U(s)=\Re e\:\mathcal{L}\{S;s\}$. Assume there are $\lambda>0$ and $g\in L^{1}(-\lambda,\lambda)$ such that If in addition $U$ has $L^{1}_{loc}$-boundary behavior on the boundary open subset $1+i(\mathbb{R}\setminus\{0\})$, namely, if there is $f\in L^{1}_{loc}(\mathbb{R}\setminus\{0\})$ such that on any finite in

Theorems & Definitions (24)

  • Theorem 1.1: Koga2021
  • Theorem 1.2: Laplace transforms
  • Theorem 1.3: Laplace-Stieltjes transforms
  • Corollary 1.4: Ingham-Karamata type theorem
  • Theorem 1.5: Power series
  • Theorem 2.1: The renewal theorem BlackwellEFP1949Kolmogorov1936
  • proof
  • Theorem 2.2: Koga2021
  • proof
  • Lemma 3.1
  • ...and 14 more