On eigenfunction expansions for a positive potential function increasing slowly to infinity

Abstract

(a is some fixed real number), where y(u) is to be L2(0, co). In this paper a corresponding convergence result is established for functions Q(U) which increase to infinity more slowly than u. Two series are said to be equiconvergent if the convergence of each series implies the convergence of the other, and to the same limit; in [4] the eigenfunction expansion off(u) at ?I = x was proved, under certain conditions on f(u), to be equiconvergent with the ordinary Fourier series off(u) at zc = X. Titchmarsh [6, Theorem 9.5, p. 1801 has p roved such an equiconvergence result under the hypothesis that f (u) is L2(0, co); his result is as follows. Let Q(U) be continuous, increasing and convex downwards. Let f(u) be L2(0, co); in the neighborhood of u = x (>O), let f (u) satisfy any condition which is sufficient for the convergence of an ordinary Fourier series (to f (x)). Then the eigenfunction expansion of.f(x) converges to the sumf(x). (A function is convex (downwards) if the graph of the function is below the chord joining any two points of the grlph.) Clearly Titchmarsh’s conditions allow p(u) to be ~1~ (c > I), exp u, exp exp u, etc. McLeod [l] has obtained a