Maximal theorems for some orthogonal series II

Abstract

If { u n } is the orthonormal sequence of eigenfunctions arising from a nonsingular (or sometimes a singular) Sturm-Liouville system with separated boundary conditions defined on (0, π), it has long been known that the eigenfunction expansion with respect to { u n } of a function φ on (0, π) has properties similar to those of the Fourier-cosine expansion of φ. For instance, there is the classical equiconvergence theorem of Haar ([5]; see [3] pp. 1616–1622 for a general survey). In this paper, by restricting attention to two specific classes of singular Sturm-Liouville systems, we shall establish a much more precise relationship between the corresponding eigenfunction expansions and the Fourier-cosine expansions. Many results for Fourier series can then be carried over to these eigenfunction expansions. The method of proof includes as special cases Fourier-Bessel functions, ultraspherical polynomials, Jacobi polynomials and any nonsingular Sturm-Liouville system.