Approximation of Smooth Functions in $L^{p(x)}_{2\pi}$ by Vallee-Poussin Means

Variable exponent $p(x)$ Lebesgue spaces $L^{p(x)}_{2\pi}$ is considered. For $f\in L^{p(x)}_{2\pi}$ Vallee–Poussin means $V_m^n(f,x)$ can be defined as $V_m^n(f,x)=\frac{1}{m+1}\sum\limits_{l=0}^mS_{n+l}(f,x),$ where $S_{k}(f,x)$ — partial Fourier sum of $f(x)$ of order $k$. Approximative properties of operators $V_m^n(f)=V_m^n(f,x)$ are investigated in $L^{p(x)}_{2\pi}$. Let $p(x)\ge1$ be $2\pi$-periodical variable exponent that satisfies Dini–Lipschitz condition. When $m=n-1$ and $m=n$ the following estimate is proved: $\|f-V_m^n(f)\|_{p(\cdot)}\le \frac{c_r(p)}{n^r}E_n(f^{(r)})_{p(\cdot)}$, where $E_n(f^{(r)})_{p(\cdot)}$ is the best approximation of function $f^{(r)}(x)$ by trigonometric polynomials of order $n$ in $L^{p(x)}_{2\pi}$.