Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Fourier series of functions of $\Lambda$-bounded variation
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by Daniel Waterman
Proc. Amer. Math. Soc. 74 (1979), 119-123
DOI: https://doi.org/10.1090/S0002-9939-1979-0521884-1

Abstract:

It is shown that the Fourier coefficients of functions of $\Lambda$-bounded variation, $\Lambda = \{ {\lambda _n}\}$, are $O({\lambda _n}/n)$. This was known for ${\lambda _n} = {n^{\beta + 1}}, - 1 \leqslant \beta < 0$. The classes L and HBV are shown to be complementary, but L and $\Lambda {\text {BV}}$ are not complementary if $\Lambda {\text {BV}}$ is not contained in HBV. The partial sums of the Fourier series of a function of harmonic bounded variation are shown to be uniformly bounded and a theorem analogous to that of Dirichlet is shown for this class of functions without recourse to the Lebesgue test.
References
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Bibliographic Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 74 (1979), 119-123
  • MSC: Primary 42A16; Secondary 42A20
  • DOI: https://doi.org/10.1090/S0002-9939-1979-0521884-1
  • MathSciNet review: 521884