Geodesic
Integral moduli of smoothness and the Fourier coefficients of the composition of functions
Sbornik. Mathematics, Tome 38 (1981) no. 4, pp. 549-561 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {A. A. Sahakian},
     title = {Integral moduli of smoothness and the {Fourier} coefficients of the composition of functions},
     journal = {Sbornik. Mathematics},
     pages = {549--561},
     year = {1981},
     volume = {38},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_38_4_a6/}
}
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A. A. Sahakian. Integral moduli of smoothness and the Fourier coefficients of the composition of functions. Sbornik. Mathematics, Tome 38 (1981) no. 4, pp. 549-561. http://geodesic.mathdoc.fr/item/SM_1981_38_4_a6/

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[3] P. L. Ulyanov, “Predstavlenie funktsii ryadami i klassy $\varphi(L)$”, Uspekhi matem. nauk, XXVII:2(164) (1972), 3–52 | MR

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