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

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 $F(x)=f(\tau(x))$.
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