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arXiv · 1204.4392

Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates

D. Gorbachev·S. Tikhonov
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Abstract

We prove two-sided inequalities between the integral moduli of smoothness of a function on $\mathbb{R}^d/\mathbb{T}^d$ and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is given by the equivalence results for functions satisfying certain regular conditions. Applications include a quantitative form of the Riemann-Lebesgue lemma as well as several other questions in approximation theory and the theory of function spaces.

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D. Gorbachev, S. Tikhonov. 2012-04-20. Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates. https://arxiv.org/abs/1204.4392

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