Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by de la Vallée-Poussin means

We consider the space $L^{p(\cdot)}_{2\pi}$ formed by $2\pi$-periodic real measurable functions $f$ for which the integral $\displaystyle\int_{-\pi}^{\pi}|f(x)|^{p(x)}\,dx$ exists and is finite, where $p(x)$, $1\leqslant p(x)$, is a $2\pi$-periodic measurable function (a variable exponent). If $p(x)\leqslant \overline p<\infty$, then the space $L^{p(\cdot)}_{2\pi}$ can be endowed with the structure of Banach space with the norm
$$ \|f\|_{p(\cdot)}=\inf\biggl\{\alpha>0:\int_{-\pi}^{\pi}\biggl|\frac{f(x)}{\alpha}\biggr|^{p(x)}\,dx\leqslant1\biggr\}. $$
In the space $L^{p(\cdot)}_{2\pi}$ we distinguish a subspace $W^{r,p(\cdot)}_{2\pi}$ of Sobolev type. We investigate the approximation properties of the de la Vallée-Poussin means for trigonometric Fourier sums for functions in the space $W^{r,p(\cdot)}_{2\pi}$. In particular, we prove that if the variable exponent $p=p(x)$ satisfies the Dini-Lipschitz condition $|p(x)-p(y)|\ln\frac{2\pi}{|x-y|}\leqslant c$ and if $f\in W^{r,p(\cdot)}_{2\pi}$, then the de la Vallée-Poussin means $V_m^n(f)=V_m^n(f,x)$ with $n\leqslant am$ satisfy
$$ \|f-V_m^n(f)\|_{p(\cdot)}\leqslant \frac{c_r(p,a)}{n^r}\Omega\biggl(f^{(r)}, \frac1n\biggr)_{p(\cdot)}, $$
where $\Omega(g,\delta)_{p(\cdot)}$ is a modulus of continuity of the function $g\in L^{p(\cdot)}_{2\pi}$ defined in terms of the Steklov functions. It is proved that if $1<p(x)$, $r\geqslant1$, $f\in W^{r,p(\cdot)}_{2\pi}$ and the Dini-Lipschitz condition holds, then
$$ |f(x)-V_m^n(f,x)|\leqslant\frac{c_r(p)}{m+1}\sum_{k=n}^{n+m}\frac{E_k(f^{(r)})_{p(\cdot)}}{(k+1)^{r-{{1}/{p(x)}}}}, $$
where $E_k(g)_{p(\cdot)}$ stands for the best approximation to $g\in L^{p(\cdot)}_{2\pi}$ by trigonometric polynomials of order $k$.
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