The surprising almost everywhere convergence of Fourier-Neumann series
Let \(J_\mu\) denote the Bessel function of the first kind of order \(\mu\) and for \(\alpha\geq-1/2\) let \[ j_n^\alpha(x):=\sqrt{2(\alpha+2n+1)}J_{\alpha+2n+1}(x)x^{-\alpha-1}. \] The system \(j_n^\alpha\) is a (non-complete) orthonormal system in \(L^2_\alpha := L^2[(0,\infty),d\mu_\alpha]\) with \(d\mu_\alpha(x):=x^{2\alpha+1}\,dx\). In \textit{Ó.~Ciaurri et. al.} [J. Math. Anal. Appl. 236, No.~1, 125--147 (1999; Zbl 0953.42015)] it is established that the Fourier series of a function \(f\) in \(L^p_\alpha\) with respect to this system (the function's Fourier-Neumann series) converges to \(M_\alpha f\) \textit{almost everywhere} whenever \(p\) is such that the Fourier-Neumann series converges to \(M_\alpha f\) \textit{in mean} in \(L^p_\alpha\). Here \(M_a\) is a certain projection in \(L^p_\alpha\). In the paper under review the latter result is in the case \(-1/2\leq\alpha<0\) extended in the sense that the Fourier-Neumann series of a function \(f\) in \(L^p_\alpha\) converges to \(M_\alpha f\) almost everywhere for the wider range of \(p\)'s such that \(4(\alpha+1)/(2\alpha+3) < p < 4(\alpha+1)/(2\alpha+1)\) (the canonical range of convergence). One can still consider Fourier-Neumann series for functions in \(L^p_\alpha\) with \(1<p\leq 4(\alpha+1)/(2\alpha+3)\), but it is an open question what the limit function is in this case.
scientific article; zbMATH DE number 5624873