Sufficient conditions for trigonometric integrals to belong to a Zygmund class of functions

Abstract

Let f: ℝ → C be a Lebesgue integrable function on the real line ℝ:= (-∞,∞) and consider its trigonometric integral defined by I(x): = ∫_ℝf(t)e^xdt, x ∈ ℝ. We give sufficient conditions in terms of certain integral means of f to ensure that I(x) belong to one of the Zygmund classes Zyg(α) and zyg(α) for some 0 < α < 2. In the particular case α = 1, our theorems are the nonperiodic versions of those of A. Zygmund on the smoothness of the sum of trigonometric series (see in [2]_and also [3, on pp. 320-321]). Our method of proof is essentially different from that used by A. Zygmund. We establish interesting interrelations between the order of magnitude of certain initial integral means and those of certain tail integral means of the function f.