Sufficient conditions for the Lebesgue integrability of Fourier transforms

Abstract

We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function f ∈ Lp(ℝ) for some 1 < p ≤ 2. These sufficient conditions are in terms of the Lp integral modulus of continuity of f; in particular, they apply for functions in the integral Lipschitz class Lip(α, p) and for functions of bounded s-variation for some 0 < s < p. Our theorems are nonperiodic versions of the classical theorems of Bernstein, Szasz, Zygmund and Salem, and recent theorems of Gogoladze and Meskhia on the absolute convergence of Fourier series.