Nikolskii inequality and functional classes on compact Lie groups

In this note we study Besov, Triebel-Lizorkin, Wiener, and Beurling function spaces on compact Lie groups. A major role in the analysis is played by the Nikolskii inequality.Comment: In this note (to appear in Funct. Anal. Appl.) we present results from our paper at arXiv:1403.3430 (to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.) in the simplified setting of compact Lie groups. We refer to the above paper for more general formulations in the setting of compact homogeneous manifolds and for the proof

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

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.Comment: 22 page