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

TL;DR

This paper establishes two-sided bounds linking the smoothness of functions to the growth of their Fourier transforms, with applications to approximation theory and function space analysis.

Contribution

It provides sharp two-sided inequalities connecting moduli of smoothness and Fourier transform tail integrals, enhancing understanding of function regularity and approximation.

Findings

Established sharp two-sided inequalities for functions on  and  torus.

Derived a quantitative form of the Riemann-Lebesgue lemma.

Applied results to approximation theory and function space analysis.

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.