Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Jackson and Stechkin type inequalities for exponantial approximation of
  functions in some non-translation invariant Banach spaces on the real line   Present work contains an elementary method to obtain Jackson and Stechkin
type inequalities of approximation by integral functions of finite degree
(IFFD) in some non-translation invariant Banach spaces of 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 $% \boldsymbol{R}$. As a model example we
consider the variable exponent Lebesgue spaces. 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 H\"{o}lder 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 $T_{\delta }$ 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 some norm in $L_{p\left( x\right) }\left( B\right) .$

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The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat