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Improper Poisson integral of the sinc function

Jerzy Szulga

TL;DR

The paper investigates the improper Poisson integral for nonintegrable functions, focusing on $f(x)=\operatorname{sinc} x$ and developing a framework around random counting measures $N(A)$ and an operator $N f=\sum_n f(S_n)$. It establishes convergence results via renewal-process techniques, LePage-style atomic representations, and modular-space criteria on $L^2_{\,\mathrm{imp}}$, showing a.s. convergence and, under moment conditions, $L^p$ convergence with unconditional behavior in the studied modes. It provides explicit analysis for a Pareto-based atomic construction, demonstrating that the auxiliary kernel $K(s)$ can be eventually integrable for sinc, and discusses extensions to other Lévy processes (e.g., $\alpha$-stable, Gamma) with varying implications for improper integration. The work also cautions about interpreting simulations of truncated sums and offers a conjecture about the a.s. summability of binned sums $\sum_n f_k(S_n)$ under suitable conditions. Overall, it advances the understanding of how nonintegrable functions can be treated within Poisson-type stochastic integrals and outlines both concrete results for sinc and avenues for further generalization.

Abstract

A formal sum $\sum_n f(S_n)$ may be seen as the integral $\int f dN$ with respect to random point process $N(A)=|\{n:S_n\in A\}|$. We study its convergence beyond the well known context of Lebesgue integrable functions, admitting nonintegrable functions whose improper Riemann-Lebesgue integrals exist. We focus on the sinc function and some of its relations leaving the general case to conjectures.

Improper Poisson integral of the sinc function

TL;DR

The paper investigates the improper Poisson integral for nonintegrable functions, focusing on and developing a framework around random counting measures and an operator . It establishes convergence results via renewal-process techniques, LePage-style atomic representations, and modular-space criteria on , showing a.s. convergence and, under moment conditions, convergence with unconditional behavior in the studied modes. It provides explicit analysis for a Pareto-based atomic construction, demonstrating that the auxiliary kernel can be eventually integrable for sinc, and discusses extensions to other Lévy processes (e.g., -stable, Gamma) with varying implications for improper integration. The work also cautions about interpreting simulations of truncated sums and offers a conjecture about the a.s. summability of binned sums under suitable conditions. Overall, it advances the understanding of how nonintegrable functions can be treated within Poisson-type stochastic integrals and outlines both concrete results for sinc and avenues for further generalization.

Abstract

A formal sum may be seen as the integral with respect to random point process . We study its convergence beyond the well known context of Lebesgue integrable functions, admitting nonintegrable functions whose improper Riemann-Lebesgue integrals exist. We focus on the sinc function and some of its relations leaving the general case to conjectures.

Paper Structure

This paper contains 9 sections, 4 theorems, 40 equations.

Table of Contents

  1. Rationale
  2. The sinc function
  3. A renewal process
  4. Alternative atomic representations
  5. LePage series
  6. Examples
  7. Discussion
  8. Other Lévy processes
  9. A potentially misleading simulation

Key Result

Theorem 2.1

Let $f(x)=\frac{e^{\imath x}}{x}$. Then series (Nf) converges a.s. If ${\mathbb E} X^{2p}<\infty$ then the series converges eventually in $L^p$ for every $p> 0$. The convergence is unconditional for each of these modes.

Theorems & Definitions (9)

  • Conjecture 1
  • Theorem 2.1
  • Lemma 2.1.1
  • Conjecture 2
  • Lemma 2.1.2
  • Theorem 2.2
  • Remark 2.2.1
  • Conjecture 3
  • Conjecture 4