Full length article: Moduli of smoothness and growth properties of Fourier transforms: Two-sided estimates

Authors:
Dmitry Gorbachev;Sergey Tikhonov
Affiliations:
Tula State University, Department of Mechanics and Mathematics, 300600 Tula, Russia;ICREA and Centre de Recerca Matemítica, Campus de Bellaterra, Edifici C - 08193 Bellaterra (Barcelona), Spain
Venue:
Journal of Approximation Theory
Year:
2012

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Abstract

We prove two-sided inequalities between the integral moduli of smoothness of a function on R^d/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.