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A partial converse to the Riemann--Lebesgue lemma for Bessel--Fourier series of order zero

Ryan L. Acosta Babb

Abstract

It is known that the Bessel--Fourier coefficients $f_m$ of a function $f$ such that $\sqrt{x}f(x)$ is integrable over $[0,1]$ satisfy $f_m/\sqrt{m}\to 0$. We show a partial converse, namely that for $0\leq α<1/2$ and any non-negative $a_m\to 0$, there is a function $f$ such that $x^{α+1}f(x)$ is integrable and its Bessel--Fourier coefficients $f_m$ satisfy $m^{-α}f_m\geq a_m$ and $m^{-α}f_m\to 0$. We conjecture that the same should be true when $α=\frac{1}{2}$, and discuss some consequences of this conjecture.

A partial converse to the Riemann--Lebesgue lemma for Bessel--Fourier series of order zero

Abstract

It is known that the Bessel--Fourier coefficients of a function such that is integrable over satisfy . We show a partial converse, namely that for and any non-negative , there is a function such that is integrable and its Bessel--Fourier coefficients satisfy and . We conjecture that the same should be true when , and discuss some consequences of this conjecture.

Paper Structure

This paper contains 9 sections, 12 theorems, 95 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. Bessel functions and their asymptotics
  4. 'Weak' decay theorems
  5. 'Strong' decay theorems
  6. Bounds on $x^{\alpha+1}K^\alpha_M(x)$
  7. Lower bounds on $\widetilde{S}^{\alpha}_M$.
  8. Bounds on $R^{\alpha}_M$.
  9. Comments for further research

Key Result

Theorem 1.1

Let $a_n\geqslant 0$ be a sequence such that $a_n\to 0$ as $\left\vert n\right\vert\to\infty$. Then, there is a function $f\in L^1([0,1])$ whose Fourier coefficients satisfy $\widehat{f}(n)\geqslant a_n$ for all $n\in\mathbb{Z}$.

Figures (3)

  • Figure 1: Plots of $K^{1/2}_M(x)$ for $0\leqslant x \leqslant 1$ with $M=10$, $M=100$ and $M=1000$, respectively. (Generated with Mathematica.)
  • Figure 2: The values of $\left\vert J_1(j_m)\right\vert$ appear monotonically decreasing.
  • Figure 3: The function $\frac{\cos{v}}{v^{1/2-\alpha}}$. The positive green area outweighs the negative red area for a positive net contribution between $y_k$ and $y_{k+1}$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark
  • Lemma 3.1: Lemma 3.3.2 in GrafakosCFA
  • Lemma 3.2: Lemma 3.3.3 in GrafakosCFA
  • ...and 14 more