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 $\boldsymbol{R}:=\left( -\infty ,+\infty \right) $. To do this, we employ a transference theorem which produce norm inequalities starting from norm inequalities in $\mathcal{C}(\boldsymbol{R})$, the class of bounded uniformly continuous functions defined on $\boldsymbol{R}$. Let $B\subseteq \boldsymbol{R}$ be a measurable set, $p\left( x\right) :B\rightarrow \lbrack 1,\infty )$ 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 $1\leq \mathop{\rm ess \; inf} \limits\nolimits_{x\in B}p(x)$, $\mathop{\rm ess \; sup}\limits\nolimits_{x\in B}p(x)<\infty $, where $I$ is the identity operator, $r\in \mathrm{N}:=\left\{ 1,2,3,\cdots \right\} $, $\delta \geq 0$ and $$ T_{\delta }f\left( x\right) =\frac{1}{\delta }\int\nolimits_{0}^{\delta }f\left( x+t\right) dt, x\in \boldsymbol{R}, T_{0}\equiv I, $$ is the forward Steklov operator. It is proved that $$ \left\Vert \left( I-T_{\delta }\right) ^{r}f\right\Vert _{p\left( \cdot \right) } $$ is a suitable measure of smoothness for functions in $L_{p\left( x\right) }\left( B\right) $, where $\left\Vert \cdot \right\Vert _{p\left( \cdot \right) }$ is Luxemburg norm in $L_{p\left( x\right) }\left( B\right) .$ We obtain main properties of difference operator $\left\Vert \left( I-T_{\delta }\right) ^{r}f\right\Vert _{p\left( \cdot \right) }$ 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( \boldsymbol{R}\right) . $

Keywords

• Variable exponent Lebesgue space
• One sided Steklov operator
• Integral functions of finite degree
• Best approximation
• Direct theorem
• Inverse theorem
• Modulus of smoothness
• Marchaud inequality
• K-functional.

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