Proceedings of the American Mathematical Society

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



Fourier series of functions of $ \Lambda $-bounded variation

Author: Daniel Waterman
Journal: Proc. Amer. Math. Soc. 74 (1979), 119-123
MSC: Primary 42A16; Secondary 42A20
MathSciNet review: 521884
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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 [Enhancements On Off] (What's this?)

  • [1] Daniel Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107–117. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity. II. MR 0310525
  • [2] -, On the summability of Fourier series of functions of $ \Lambda $-bounded variation, Studia Math. 55 (1976), 87-95.
  • [3] Daniel Waterman, On 𝐿-bounded variation, Studia Math. 57 (1976), no. 1, 33–45. MR 0417355
  • [4] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

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Article copyright: © Copyright 1979 American Mathematical Society