On the magnitude of Fourier coefficients
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- by Michael Schramm and Daniel Waterman PDF
- Proc. Amer. Math. Soc. 85 (1982), 407-410 Request permission
Abstract:
If $f$ is a function on ${R^1}$ of $\Lambda$-bounded variation and period $2\pi$, then its $n$th Fourier coefficient $\hat f(n) = O(1/\Sigma _1^n1/{\lambda _j})$ and its integral modulus of continuity ${\omega _1}(f;\delta ) = O(1/\Sigma _1^{[1/\delta ]}1/{\lambda _j})$. The result on $\hat f(n)$ is best possible in a sense. These results can be extended to certain other classes of functions of generalized variation.References
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- Masaaki Shiba, On absolute convergence of Fourier series of function of class $\Lambda -\textrm {BV}^{(p)}$, Sci. Rep. Fac. Ed. Fukushima Univ. 30 (1980), 7–10. MR 599729 S. Wang, Personal communication.
- Daniel Waterman, On the summability of Fourier series of functions of $L$-bounded variation, Studia Math. 54 (1975/76), no. 1, 87–95. MR 402391, DOI 10.4064/sm-55-1-87-95
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- Daniel Waterman, Fourier series of functions of $\Lambda$-bounded variation, Proc. Amer. Math. Soc. 74 (1979), no. 1, 119–123. MR 521884, DOI 10.1090/S0002-9939-1979-0521884-1
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 85 (1982), 407-410
- MSC: Primary 42A16; Secondary 26A45
- DOI: https://doi.org/10.1090/S0002-9939-1982-0656113-1
- MathSciNet review: 656113