On the degree of convergence of expansions in an infinite interval
William Edmund Milne · Transactions of the American Mathematical Society · 1929
In a recent investigation t of the degree of convergence of the Gram-Charlier series in the infinite interval -oo <a;<co the writer has shown that in general the convergence is very much less rapid than in the case of a Fourier series under similar circumstances.The question arises whether the slow rate of convergence of this particular series is a characteristic of all such expansions on an infinite interval, or whether there may exist series of orthogonal functions for which the rate of convergence is as rapid as that of the Fourier series in a finite interval.A study of this question not only is of some theoretical interest, but may conceivably serve a very practical end by leading to the discovery of series better adapted to the representation of frequency functions than is the slowly convergent Gram-Charlier series.This paper is devoted to those expansions on the infinite interval which are associated with the differential equation (1) d2u/dx2+ [X -qix)]u = 0, in which qix) is real and continuous for all values of x, and(2) lim qix) = + oo .x=± » The differential equation above is a special case of equations investigated on the infinite interval by Weyl,J Hilb, § Gray,|| and Milne.If It has been shown that there exists an infinite set of critical values of X, X0, Xi, X2, • • • with limit point at + oo only, corresponding to which equation (1) has solutions I/o0*0, Uiix), U2ix), • ■ ■ , satisfying the conditions (3) lim U"ix) = lim Uiix) = 0. X=± « X=± to