Approximation of functions in $L^{p(x)}_{2\pi}$ by trigonometric polynomials

We consider the Lebesgue space $L^{p(x)}_{2\pi}$ with variable exponent $p(x)$. It consists of measurable functions $f(x)$ for which the integral $\int_0^{2\pi}|f(x)|^{p(x)}\,dx$ exists. We establish an analogue of Jackson's first theorem in the case when the $2\pi$-periodic variable exponent $p(x)\geqslant1$ satisfies the condition
\begin{equation*} |p(x')-p(x'')|\ln\frac{2\pi}{|x'-x''|}=O(1),\qquad x',x''\in[-\pi,\pi]. \end{equation*}
Under the additional assumption $p_-=\min_x p(x)>1$ we also get an analogue of Jackson's second theorem. We establish an $L^{p(x)}_{2\pi}$-analogue of Bernstein's estimate for the derivative of a trigonometric polynomial and use it to prove an inverse theorem for the analogues of the Lipschitz classes $\mathrm{Lip}(\alpha,M)_{p(\,\cdot\,)}\subset L^{p(x)}_{2\pi}$ for $0<\alpha<1$. Thus we establish direct and inverse theorems of the theory of approximation by trigonometric polynomials in the classes $\mathrm{Lip}(\alpha,M)_{p(\,\cdot\,)}$. In the definition of the modulus of continuity of a function $f(x)\in L^{p(x)}_{2\pi}$, we replace the ordinary shift $f^h(x)=f(x+h)$ by an averaged shift determined by Steklov's function $s_h(f)(x)=\frac{1}{h}\int_0^hf(x+t)\,dt$.