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Ramazan Akgün

Publications and source records attributed to Ramazan Akgün.

6 recordsLinked to original sources
arXiv2023

A transference result for Lebesgue spaces with $A_{\infty }$ weights and its applications

In this work we obtain a transference theorem for Lebesgue spaces with $A_{\infty }$ weights, namely, starting from some uniform-norm inequalities it is possible to obtain similar inequalities in Lebesgue spaces with $A_{\infty }$ weights. This transference technic allows us to obtain some weighted norm inequalities easily. Also transference result gives possibility to use fractional difference operators in weighted Lebesgue spaces easier than the classical known one. We can obtain some norm-like inequalities easily as a consequence. Some important approximation inequalities of approximation by integral functions of finite degree can be obtained with a different proof.

math.FA↗
arXiv2022

Exponential approximation in variable exponent Lebesgue spaces on the real line

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_{δ}\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\} $, $δ\geq 0$ 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 $\left\Vert \left( I-T_{δ}\right) ^{r}f\right\Vert _{p\left( \cdot \right) }$ 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). $

math.FA↗
arXiv2022

Exponential approximation of functions in Lebesgue spaces with Muckenhoupt weight

Using a transference result, several inequalities of approximation by entire functions of exponential type in $\mathcal{C}(\mathbf{R})$, the class of bounded uniformly continuous functions defined on $\mathbf{R}:=\left( -\infty ,+\infty \right) $, are extended to the Lebesgue spaces $L^{p}\left( \mathbf{\varrho }dx\right) $ $1\leq p<\infty $ with Muckenhoupt weight $\mathbf{\varrho }$ ($1\leq p<\infty $). This gives us a different proof of Jackson type direct theorems and Bernstein-Timan type inverse estimates in $L^{p}\left( \mathbf{\varrho }dx\right) $. Results also cover the case $p=1$.

math.CA↗
arXiv2021

Jackson-Stechkin type inequalities for differentiable functions in weighted Orlicz spaces

In the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt's $A_{p}$ condition. To obtain refined version of the Jackson type inequality we prove an extrapolation theorem, Marcinkiewicz multiplier theorem and Littlewood Paley type results. As a consequence refined inverse Marchaud type inequalities are obtained. By means of a realization result we find an equivalence between the fractional order weighted modulus of smoothness and the classical weighted Peetre's K-functional.

math.CA↗
arXiv2021

Weighted norm inequalities in Lebesgue spaces with Muckenhoupt weights and some applications to operators

In the present work we give a simple method to obtain weighted norm inequalities in Lebesgue spaces $L_{p,γ}$ with Muckenhoupt weights $γ$. This method is different from celebrated Extrapolation or Interpolation Theory. In this method starting point is uniform norm estimates of special form. Then a procedure give desired weighted norm inequalities in $L_{p,γ}.$ We apply this method to obtain several convolution type inequalities. As an application we consider a difference operator of type $Δ_{v}^{r}:=\left( \mathbb{I}-\mathfrak{T}_{v}\right) ^{r}$ where $\mathbb{I}$ is the identity operator, $r\in \mathbb{N}$ and \begin{equation*} \mathfrak{T}_{v}f\left( x\right) :=\frac{1}{v}\int\nolimits_{x}^{x+v}f\left(t\right) dt,\quad x\in \left[ -π,π\right] ,\quad v>0,\quad \mathfrak{T}_{0}:=\mathbb{I}. \end{equation*} We obtain main properties of $Δ_{v}^{r}f$ for functions $f$ given in $L_{p,γ}$, $1\leq p<\infty $, with weights $γ$ satisfying the Muckenhoupt's $A_{p}$ condition. Also we consider some applications of difference operator $Δ_{v}^{r}$ in these spaces. In particular, we obtain that difference $\left\Vert Δ_{v}^{r}f\right\Vert _{p,γ}$ is a useful tool for computing the smoothness properties of functions these spaces. It is obtained that $\left\Vert Δ_{v}^{r}f\right\Vert_{p,γ}$ is equivalent to Peetre's K-functional.

math.CA↗