On classes of summable functions and their Fourier Series

1. Functions which are summable may be such that certain functions of them are themselves summable. When this is the case they will possess certain special properties additional to those which the mere summability involves. A remarkable instance where this has been recognised is in the case of summable functions whose squares also are summable. The—in its formal statement almost self-evident—Theorem of Parseval which asserts that the sum of the squares of the coefficients of a Fourier series of a function f ( x ) is equal to the integral of the square of f ( x ), taken between suitable limits and multiplied by a suitable constant, has been recognised as true for all functions whose squares are summable. Moreover, not only has the converse of this been shown to be true, but writers have been led to develop a whole theory of this class of functions, in connection more especially with what are known as integral equations. That functions whose (1 + p )th power is summable, where p >0, but is not necessarily unity, should next be considered, was, of course, inevitable. As was to be expected, it was rather the integrals of such functions than the functions themselves whose properties were required. Lebesgue had already given the necessary and sufficient condition that a function should be an integral of a summable function. F. Riesz then showed that the necessary and sufficient condition that a function should be the integral of a function whose (1 + p )th power is summable had a form which constituted rather the generalisation of tire expression of the fact that such a function has bounded variation, than one which included the condition of Lebesgue as a particular case.