On the Solution of a General Transform.

6 This relation defines q for the rest of the paper, also. 7A more detailed proof will be included in the paper referred to in footnote 2. 8 Banach, S., Th6orie des operations lin&aires, Warsaw, 1932, p. 57; "Lemme." 9 Riesz, F., "Untersuchungen uber Systeme integrierbarer Funktionen," Math. Ann., 69,449-497 (1910); p. 475. ' o See Titchmarsh, E. C., Theory of Fourier Integrals, Oxford (1937), p. 105. "I Wiener, N., The Fourier Integral and Certain of Its Applications, Cambridge, England (1933), pp. 49-50. 1 The spectrum of a function of bounded variation consists of all points such that the total variation of the function is positive over every neighborhood of the point. 18 Salemn, R., "On Singular Monotonic Functions of the Cantor Type," Jour. Math, and Phys., 21, 69-82 (1942).14 Weil, A., "L'Integration dans les groupes topologiques et ses applications," Act. Sc. et Ind., No. 869, Paris, p. 55 (1940). 15 We are pleased to thank J. D. Tamarkin and A. Zygmund for suggesting a modification in the statement of Lemma 2 which results in a shortening of this part of our proof. 16 It seems noteworthy that this argument holds for p = 2 also, and together with the proof for Case -1, provides a new and somewhat simpler proof of Wiener's theorem for the space L2.