Divergent Laguerre series (Q2701589)

Fix \(a\) in \((-1,\infty)\), and consider the Laguerre functions \({\mathcal L}^a_n\), defined by NEWLINE\[NEWLINE {\mathcal L}^\alpha_n(x) = (\Gamma(n+1)/\Gamma(n+a+1))^{1/2} e^{-x/2} x^{a/2} L^a_n(x), NEWLINE\]NEWLINE where \(L^a_n\) is the usual Laguerre polynomial. For \(f\) in \(L^p(0,+\infty)\) (relative to Lebesgue measure), the Laguerre series of \(f\) is given by \(\sum_{n=0}^{+\infty}c_n {\mathcal L}^a_n\), where \(c_n = \int_0^{+\infty} f(x) {\mathcal L}^a_n(x) dx\). It is shown that, if \(4 < p \leq + \infty\), then there exists \(f\) in \(L^p(0, +\infty)\) whose Laguerre series diverges almost everywhere.