Comparison of the Pr-integral with Burkill's integrals and some applications to trigonometric series

Abstract

It is proved that the Pr-integral [9] which recovers a function from its derivative defined in the space Lr, 1 <= r < infinity, is properly included in Burkill's trigonometric CP-and SCP-integrals. As an application to harmonic analysis, a de La Vallee-Poussin-type theorem for the Pr-integral is obtained: convergence nearly everywhere of a trigonometric series to a Pr-integrable function f implies that this series is the Pr-Fourier series of f.(c) 2023 Elsevier Inc. All rights reserved.