Exponential approximation in variable exponent Lebesgue spaces on the real line
نویسندگان
چکیده
Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions finite degree (IFFD) in some variable exponent Lebesgue space real defined on $\boldsymbol{R}:=\left( -\infty ,+\infty \right) $. To do this, we employ transference theorem which produce norm starting from $\mathcal{C}(\boldsymbol{R})$, the class bounded uniformly continuous $\boldsymbol{R}$. Let $B\subseteq \boldsymbol{R}$ be measurable set, $p\left( x\right) :B\rightarrow \lbrack 1,\infty )$ function. For $f$ belonging spaces $L_{p\left( }\left( B\right) $, consider difference operator $\left( I-T_{\delta }\right)^{r}f\left( \cdot $ under condition that $p(x)$ satisfies log-Hölder continuity $1\leq \mathop{\rm ess \; inf} \limits\nolimits_{x\in B}p(x)$, $\mathop{\rm sup}\limits\nolimits_{x\in B}p(x)<\infty where $I$ is identity operator, $r\in \mathrm{N}:=\left\{ 1,2,3,\cdots \right\} $\delta \geq 0$ $$ T_{\delta }f\left( =\frac{1}{\delta }\int\nolimits_{0}^{\delta x+t\right) dt, x\in \boldsymbol{R}, T_{0}\equiv I, forward Steklov operator. It proved \left\Vert \left( }\right) ^{r}f\right\Vert _{p\left( } suitable measure smoothness for $\left\Vert \right\Vert }$ Luxemburg .$ We main properties give proof direct inverse theorems IFFD \boldsymbol{R}\right) .
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ژورنال
عنوان ژورنال: Constructive mathematical analysis
سال: 2022
ISSN: ['2651-2939']
DOI: https://doi.org/10.33205/cma.1167459