Fourier series of functions whose Hankel transform is supported on \([0,1]\)
Denote by \(J_ \mu(x)\) the Bessel function of order \(\mu\). As it is known, for \(\alpha> -1\), the system \(\{x^{-\alpha/2-1/2} J_{\alpha+ 2n+1} (x^{1/2})\): \(n= 0,1,\dots\}\) is orthogonal on \(L^ 2((0,\infty),x^ \alpha dx)\). The present author studies the mean convergence of Fourier series with respect to this system for functions whose Hankel transform is supported on \([0,1]\). The following theorem is typical. Let \(1< p<\infty\) and \(\alpha\geq -1/2\). Then there exists a constant \(C\) independent of \(n\) and \(f\) such that \(\| S_ n f\|_{L^ p(x^ \alpha)}\leq C\| f\|_{L^ p(x^ \alpha)}\) for all \(f\in L^ p(x^ \alpha):= L^ p((0,\infty),x^ \alpha dx)\) if and only if \(\max\{p_ 0,4/3\}< p< \min\{p_ 1,4\}\), where \(S_ n f\) is the \(n\)th partial sum of the Fourier series of \(f\), \(p_ 0:= {4(\alpha+1)\over 2\alpha+3}\) and \(p_ 1:= {4(\alpha+1)\over 2\alpha+1}\).
scientific article; zbMATH DE number 563356