Fourier's series and analytic functions

Abstract

Abstract 1.1. In the theorems which follow we are concerned with functions f(x) real for real x and integrable in the sense of Lebesgue. We do not, however, remain in the field of the real variable, for we suppose, in 4 et seq., that f(x), or a function associated with f(x), is analytic, or, at any rate, harmonic, in a region of the complex plane associated with the particular real value of x considered. The Fourier’s series considered are those associated with the interval (0, 2π). If a is a point of the interval, we write ϕ(u) = ½ {f (a + u) + f (a - u) - 2s} (0 < a < 2π), (1.11) ϕ(u) = ½ {f (u) + f (2π - u) - 2s} (a = 0, a = 2π), (1.12) where s is a constant.