Representation of Fourier Integrals as Sums

Here sn x is an abbreviation for sin (wx/2). This paper gives other conditions for the validity of these identities. The previous conditions permitted to have various types of discontinuity. The present paper is concerned with smooth functions; however, the growth at 0 and «> is permitted to be greater than before. Theorems 1, 2, and 3 of the previous paper together with Theorems 2 and 3 of this paper form a fairly complete elementary theory of these identities. The proofs given here do not depend on the previous paper. The results of this paper hinge on the possibility of defining the Fourier sine transform for functions which do not vanish at infinity. Theorem 1 below shows that this is possible merely by employing summability. It is to be noted that Theorem 1 is not true, as it stands, for the cosine transform. For example, the cosine transform of any constant evaluated by such a definition would vanish. Hence the inversion formula could not apply. A theory of "generalized Fourier