Some generalizations of contraction theorems for Fourier series
Let \(f\in L_{2\pi}\), \(f\sim \sum^{\infty}_{n=-\infty}c_ ne^{inx}\), \(0<p\leq 2\), and let I(t) be a positive, increasing function in (0,1) satisfying some inequalities. After generalizing B. S. Yadov's theorem on absolute convergence with power p of the series of Fourier coefficients of the shrivel contraction of f, there are given by means of the sequence \(\{c_ n\}\) necessary and sufficient conditions in order that \(A_{p,j,I}(f)=\{\int^{1}_{0}[\int^{2\pi}_{0}| \Delta^ j\!_ tf(x)I^{-1}(t)|^ 2dx]^{p/2}\frac{dt}{t}\}^{1/p}<\infty.\) where \(\Delta^ j\!_ tf(x)\) is the ordinary difference of f of order j.
scientific article; zbMATH DE number 3846169