The Lebesgue constants for Fourier-Bessel series, mean convergence, and the order of singularity of a Sturm-Liouville equation

Abstract

Questions of mean convergence of classical orthogonal expansions and rates of divergence of their Lebesgue constants are dealt with, under two aspects. The first aspect is that the known results for Jacobi, Laguerre, and Fourier-Bessel series can be seen to be closely related to each other with respect to the kind of singularities in their Sturm-Liouville equations. The second, and the main aspect is to show that the rate of divergence of the Lebesgue constants for Fourier-Bessel series, which were unknown so far, fits well into this interpretation. For the latter purpose we use the Hankel translation in order to reduce the kernel of the Fourier-Bessel partial sum to a function of one variable, a representation of which is derived by the residue calculus. This method of proof is also discussed in connection with the methods used for the other orthogonal systems and with possible generalizations to more general eigenfunction expansions.