Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by finite Fourier-Haar series

The paper deals with the space $L^{p(x)}$ consisting of classes of real measurable functions $f(x)$ on $[0,1]$ with finite integral $\displaystyle\int_0^1|f(x)|^{p(x)}\,dx$. If $1\le p(x)\le \overline p<\infty$, then the space $L^{p(x)}$ can be made into a Banach space with the norm $\displaystyle\|f\|_{p(\cdot)}=\inf\biggl\{\alpha\,{>}\,0: \int_0^1 |{f(x)/\alpha}|^{p(x)}\,dx\le\nobreak 1\biggr\}$. The inequality $\|f-Q_n(f)\|_{p(\cdot)}\le c(p)\Omega(f,1/n)_{p(\cdot)}$, which is an analogue of the first Jackson theorem, is shown to hold for the finite Fourier-Haar series $Q_n(f)$, provided that the variable exponent $p(x)$ satisfies the condition $|p(x)-p(y)|\ln(1/|x-y|)\le\nobreak c$. Here, $\Omega(f,\delta)_{p(\cdot)}$ is the modulus of continuity in $L^{p(x)}$ defined in terms of Steklov functions. If the function $f(x)$ lies in the Sobolev space $W_{p(\cdot)}^1$ with variable exponent $p(x)$, it is shown that $\|f-Q_n(f)\|_{p(\cdot)}\le c(p)/n\|f'\|_{p(\cdot)}$. Methods for estimating the deviation $|f(x)-Q_n(f,x)|$ for $f(x) \in W_{p(\cdot)}^1$ at a given point $x \in [0,1]$ are also examined. The value of $\sup_{f\in W_{p}^1(1) }|f(x)-Q_n(f,x)|$ is calculated in the case when $p(x) \equiv p = \nobreak \mathrm{const}$, where $W_{p}^1(1)=\{f\in W_{p}^1:\|f'\|_{p(\cdot)}\le1\}$.
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