Integral moduli of smoothness and the Fourier coefficients of the composition of functions

A. A. Sahakian

Sbornik. Mathematics, Tome 38 (1981) no. 4, pp. 549-561

Résumé

Using the integral modulus of smoothness, estimates for the Fourier coefficients of a composition of functions are obtained in this paper. It is proved, for example, that for any function $f(x)\in C(0,2\pi)$ and any positive sequence $\{\varepsilon_n\}_{n=1}^\infty$ with $$ 1=\varepsilon_1\geqslant\varepsilon_2\geqslant\dotsb,\qquad\sum_{n=1}^\infty\frac{\varepsilon_n}n=\infty $$ there exists a monotone continuous function $\tau(x)$ ($\tau(0)=0$, $\tau(2\pi)=2\pi$) such that $$ |a_n(F)|+|b_n(F)|= O(\varepsilon_n n^{-1}+n^{-3/2}), $$ where $a_n(F)$ and $b_n(F)$ are the Fourier coefficients of the function $F(x)=f(\tau(x))$. Bibliography: 4 titles.