Supplement to F. Wiener sieve theorem

Abstract

The theorem of F. Wiener about sieve states: the norm of the operator $\mathcal W_{r,s} : f(z)=\sum_{j=0}^\infty \widehat f_jz^j\mapsto \sum_{j=0}^\infty \widehat f_{js+r}z^j$ on the space of bounded holomorphic in the disk $\mathbb D=\{z : |z|<1\}$ functions is 1 if $r<s$. We prove that restriction $r<s$ is a final condition in this theorem. We give an applications of the Wiener's theorem to the problems of the best approximations of functions in the Hardy space $H_p$.