Fourier series of functions of $\Lambda$-bounded variation
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- by Daniel Waterman PDF
- Proc. Amer. Math. Soc. 74 (1979), 119-123 Request permission
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
- Daniel Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107–117. MR 310525, DOI 10.4064/sm-44-2-107-117 —, On the summability of Fourier series of functions of $\Lambda$-bounded variation, Studia Math. 55 (1976), 87-95.
- Daniel Waterman, On $L$-bounded variation, Studia Math. 57 (1976), no. 1, 33–45. MR 417355, DOI 10.4064/sm-57-1-33-45
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional 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