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In a recent investigation t of the degree of convergence of the GramCharlier 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

On the degree of convergence of expansions in an infinite interval