On the magnitude of Fourier coefficients

Schramm, Michael; Waterman, Daniel

doi:10.1090/S0002-9939-1982-0656113-1

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.

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