Exponential approximation in variable exponent Lebesgue spaces on the real line

Authors : Ramazan AKGÜN

Pages : 214-237

Doi:10.33205/cma.1167459

View : 11 | Download : 8

Publication Date : 2022-12-01

Article Type : Research Paper

Abstract :Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions of finite degree insert ignore into journalissuearticles values(IFFD); in some variable exponent Lebesgue space of real functions defined on $\boldsymbol{R}:=\leftinsert ignore into journalissuearticles values( -\infty ,+\infty \right); $. To do this, we employ a transference theorem which produce norm inequalities starting from norm inequalities in $\mathcal{C}insert ignore into journalissuearticles values(\boldsymbol{R});$, the class of bounded uniformly continuous functions defined on $\boldsymbol{R}$. Let $B\subseteq \boldsymbol{R}$ be a measurable set, $p\leftinsert ignore into journalissuearticles values( x\right); :B\rightarrow \lbrack 1,\infty );$ be a measurable function. For the class of functions $f$ belonging to variable exponent Lebesgue spaces $L_{p\leftinsert ignore into journalissuearticles values( x\right); }\leftinsert ignore into journalissuearticles values( B\right); $, we consider difference operator $\leftinsert ignore into journalissuearticles values( I-T_{\delta }\right);^{r}f\leftinsert ignore into journalissuearticles values( \cdot \right); $ under the condition that $pinsert ignore into journalissuearticles values(x);$ satisfies the log-Hölder continuity condition and $1\leq \mathop{\rm ess \; inf} \limits\nolimits_{x\in B}pinsert ignore into journalissuearticles values(x);$, $\mathop{\rm ess \; sup}\limits\nolimits_{x\in B}pinsert ignore into journalissuearticles values(x);<\infty $, where $I$ is the identity operator, $r\in \mathrm{N}:=\left\{ 1,2,3,\cdots \right\} $, $\delta \geq 0$ and $$ T_{\delta }f\leftinsert ignore into journalissuearticles values( x\right); =\frac{1}{\delta }\int\nolimits_{0}^{\delta }f\leftinsert ignore into journalissuearticles values( x+t\right); dt, x\in \boldsymbol{R}, T_{0}\equiv I, $$ is the forward Steklov operator. It is proved that $$ \left\Vert \leftinsert ignore into journalissuearticles values( I-T_{\delta }\right); ^{r}f\right\Vert _{p\leftinsert ignore into journalissuearticles values( \cdot \right); } $$ is a suitable measure of smoothness for functions in $L_{p\leftinsert ignore into journalissuearticles values( x\right); }\leftinsert ignore into journalissuearticles values( B\right); $, where $\left\Vert \cdot \right\Vert _{p\leftinsert ignore into journalissuearticles values( \cdot \right); }$ is Luxemburg norm in $L_{p\leftinsert ignore into journalissuearticles values( x\right); }\leftinsert ignore into journalissuearticles values( B\right); .$ We obtain main properties of difference operator $\left\Vert \leftinsert ignore into journalissuearticles values( I-T_{\delta }\right); ^{r}f\right\Vert _{p\leftinsert ignore into journalissuearticles values( \cdot \right); }$ in $L_{p\leftinsert ignore into journalissuearticles values( x\right); }\leftinsert ignore into journalissuearticles values( B\right); .$ We give proof of direct and inverse theorems of approximation by IFFD in $L_{p\leftinsert ignore into journalissuearticles values( x\right); }\leftinsert ignore into journalissuearticles values( \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