Ryan Acosta Babb: 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 \(x^{s+1/2}f(x)\) is integrable over \([0,1]\) with \(0\leq s \leq \nu+1/2\), satisfy \(j_{\nu,m}^{-s-1/2}f_{\nu,m}\to 0\).

We show a partial converse, namely, that for \(0\leq \alpha<1/2\) and any non-negative \(c_m\to 0\), there is a function \(f\) such that \(x^{\alpha+1+\nu}f(x)\) is integrable and its Bessel--Fourier coefficients \(f_{\nu,m}\) satisfy \(j_{\nu,m}^{-\alpha}f_{\nu,m}\geq c_m\) 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.