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

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 $f_m\mathrm{/}\sqrt{m}\to 0$. We show a partial converse, namely that for $0\le \alpha <1\mathrm{/}2$ 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 $m^{-\alpha }{f}_m\ge a_m$ 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.