IV. Sturm-Liouville series of normal functions in the theory of integral equations

One of the most important branches of the theory of integral equations is connected with the problem of representing a function as a series of normal functions, Hilbert and Schmidt, who made the earliest contributions, have been able to obtain sufficient conditions under which an assigned function may be expanded in terms of a system of normal functions belonging to a symmetric characteristic function (kern). These conditions are narrow in respect to the nature of the function which may be expanded, but they have the advantage of applying to very general systems of normal functions. They apply, in particular, to the expansion of a function in both the sine and cosine series of Fourier. It is in the light of our knowledge of the properties of the latter series that the narrowness above referred to becomes evident. In point of fact, the Hilbert-Schmidt theory is only applicable to Fourier’s series corresponding to a function which has a continuous second differential coefficient in (0, π), and which furthermore satisfies certain boundary conditions at the end points of the interval. For example, in the case of the sine series, the function must vanish at both end points. It would appear, therefore, that the wide generality as to the system of normal functions is obtained at the cost of the generality of the function which it is desired to represent. Later memoirs by Kneser and Hobson have made it abundantly clear that, by restricting the nature of the system of normal functions, results may be obtained in regard to the representation of very much wider classes of functions than were contemplated by Hilbert and Schmidt. Kneser’s paper is of importance as marking the first step in this direction, hut his results are far less general than those obtained by Hobson and published last year in the ‘Proceedings of the London Mathematical Society.’ As one of many interesting applications of a general convergence theorem, the latter has been able to show that any Sturm-Liouville series corresponding to an assigned function converges at a point, provided that the function has a Lebesgue integral in the interval of representation, and is of limited total fluctuation in an arbitrarily small neighbourhood of the point in question. Taken in conjunction with other results of a similar kind, this cannot fail to suggest the possibility of extending most of the well-known theorems on Fourier’s series to the whole class of Sturm-Liouville expansions. It is the purpose of this memoir to show that all the more important theorems are capable of this extension.