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

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