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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
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.


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Article copyright: © Copyright 1979 American Mathematical Society