Wiener's positive Fourier coefficients theorem in variants of Lp spaces

Abstract

In this paper we consider spaces that are “close” to L(T): L itself; the space of functions f with positive Fourier coefficients that have |f |p integrable near 0; the space of functions whose Fourier coefficients are in p ; the space of functions whose Fourier coefficients {cn} satisfy ∑|cn|pnp−2 < ∞; and the mixed norm spaces p ,2, 1 < p < 2. We shall describe several relationships between these spaces. Let T be the interval [−π,π]. For every 1 ≤ p < ∞, we say that a measurable function f is in L = L(T) if ‖f ‖p p = 1 2π ∫