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Hankel transforms of general monotone functions

Alberto Debernardi

Abstract

We show that the Hankel transform of a general monotone function converges uniformly if and only if the limit function is bounded. To this end, we rely on an Abel-Olivier test for real-valued functions. Analogous results for cosine series are derived as well. We also show that our statements do not hold without the general monotonicity assumption in the case of cosine integrals and series.

Hankel transforms of general monotone functions

Abstract

We show that the Hankel transform of a general monotone function converges uniformly if and only if the limit function is bounded. To this end, we rely on an Abel-Olivier test for real-valued functions. Analogous results for cosine series are derived as well. We also show that our statements do not hold without the general monotonicity assumption in the case of cosine integrals and series.
Paper Structure (4 sections, 12 theorems, 74 equations)

This paper contains 4 sections, 12 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Bessel functions
  3. Abel-Olivier test for $GM$ functions and sequences
  4. Proofs

Key Result

Theorem 1.2

Let $f\in GM$ be real-valued. If the integral converges, then

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem A
  • Theorem 1.6
  • Remark 2.1
  • Lemma 2.2
  • proof
  • ...and 16 more