It is proved that the $P_r$-integral [9] which recovers a function from its derivative defined in the space $L^r$, 1 ≤r<∞, is properly included in Burkill’s trigonometric CP-and SCP-integrals. As an application to harmonic analysis, a de La Vallée-Poussin-type theorem for the $P_r$-integral is obtained: convergence nearly everywhere of a trigonometric series to a $P_r$-integrable function f implies that this series is the Pr-Fourier series of f.
Musial, P., Skvortsov, V., Tulone, F. (2023). Comparison of the $P_r$-integral with Burkill's integrals and some applications to trigonometric series. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 523(1) [10.1016/j.jmaa.2023.127019].
Comparison of the $P_r$-integral with Burkill's integrals and some applications to trigonometric series
Tulone, Francesco
2023-01-01
Abstract
It is proved that the $P_r$-integral [9] which recovers a function from its derivative defined in the space $L^r$, 1 ≤r<∞, is properly included in Burkill’s trigonometric CP-and SCP-integrals. As an application to harmonic analysis, a de La Vallée-Poussin-type theorem for the $P_r$-integral is obtained: convergence nearly everywhere of a trigonometric series to a $P_r$-integrable function f implies that this series is the Pr-Fourier series of f.
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