Exponential approximation in variable exponent Lebesgue spaces on the real line

Abstract

Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions of finite degree (IFFD) in some variable exponent Lebesgue space of real functions defined on $\mathbf{R}:=\left(-\infty ,+\infty \right)$. To do this we employ a transference theorem which produce norm inequalities starting from norm inequalities in $\mathcal{C}(\mathbf{R})$, the class of bounded uniformly continuous functions defined on $\mathbf{R}$. Let $B\subseteq \mathbf{R}$ be a measurable set, $p\left(x\right):B\to [1,\infty )$ be a measurable function. For the class of functions $f$ belonging to variable exponent Lebesgue spaces $L_{p\left(x\right)}\left(B\right)$ we consider difference operator ${\left(I-T_{\delta }\right)}^{r}f\left(\cdot \right)$ under the condition that $p(x)$ satisfies the Log Hölder continuity condition and $1\le {\mathrm{essinf}}_{x\in B}p(x)$, ${\mathrm{esssup}}_{x\in B}p(x)<\infty$ where $I$ is the identity operator, $r\in \mathrm{N}:=\left\{1,2,3,\cdots \right\}$, $\delta \ge 0$ and \begin{equation*} T_{δ}f\left( x\right) =\frac{1}{δ}\int\nolimits_{0}^{δ}f\left( x+t\right) dt ) \end{equation*} is the forward Steklov operator. We obtain main properties of difference operator ${\Vert {\left(I-T_{\delta }\right)}^{r}f\Vert }_{p\left(\cdot \right)}$ in $L_{p\left(x\right)}\left(B\right).$ We give proof of direct and inverse theorems of approximation by IFFD in $L_{p\left(x\right)}\left(\mathbf{R}\right).$

arXiv comment26 pages, submitted