On the rate of growth of Bessel–Fourier coefficients for integrable functions

Abstract

It is known that the Bessel–Fourier coefficients $f_{\nu ,m}$ of a function f, such that $x^{s+1/2}f\left(x\right)$ is integrable over $[0,1]$ with $0⩽s⩽\nu +1/2$, satisfy $j_{\nu ,m}^{-s-1/2}{f}_{\nu ,m}\to 0$. We show a partial converse, namely that for $0⩽\alpha <1/2$ and any non-negative $a_m\to 0$, there is a function f such that $x^{\alpha +1+\nu }f\left(x\right)$ is integrable and its Bessel–Fourier coefficients $f_{\nu ,m}$ satisfy $j_{\nu ,m}^{-\alpha }{f}_{\nu ,m}⩾a_m$ and $j_{\nu ,m}^{-\alpha }{f}_{\nu ,m}\to 0$. For $\nu =0$, we conjecture that the same should be true when $\alpha =\frac{1}{2}$, and discuss some consequences of this conjecture for the divergence Bessel–Fourier expansions of radial functions on the disc.