Jackson and Stechkin type inequalities of trigonometric approximation inAp;q()w;#

Avşar, Ahmet Hamdi; Koç, Hüseyin

doi:10.3906/mat-1712-84

Cited by 2 publications

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“…See the books [16,18,51] for more references. Nowadays, many mathematician solved many problems for the approximation of function in these type spaces defined on [0, 2π] ⊂ R (see e.g., [7,8,26,30,31,34], [1,2,3,11,12], [5,6,9,13,14], [22,24,25,28,32,33,36], [37,38,44,49,50,56]). In this paper, we propose generalized our last results in [10] which we obtained a direct and inverse theorems for approximation by entire functions of finite degree in variable exponent Lebesgue spaces on the whole real axis R with (1.1) sup…”

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confidence: 99%

Exponential approximation in variable exponent Lebesgue spaces on the real line

Akgün

2022

Constructive Mathematical Analysis

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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) . $

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