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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
DOI: https://doi.org/10.1090/S0002-9939-1979-0521884-1
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] D. Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107-117; errata, ibid. 44 (1972), 651. MR 0310525 (46:9623)
  • [2] -, On the summability of Fourier series of functions of $ \Lambda $-bounded variation, Studia Math. 55 (1976), 87-95.
  • [3] -, On $ \Lambda $-bounded variation, Studia Math. 57 (1976), 33-45. MR 0417355 (54:5408)
  • [4] A. Zygmund, Trigonometric series, Vol. I, Cambridge Univ. Press, Cambridge, 1959. MR 0107776 (21:6498)

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DOI: https://doi.org/10.1090/S0002-9939-1979-0521884-1
Article copyright: © Copyright 1979 American Mathematical Society

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