Inverse Theorems for Functions in L p and other Spaces

Ditzian, Z.

doi:10.2307/2040754

Cited by 2 publications

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“…Some inequalities of Approximation Theory in a Homogenous Banach Spaces (HBS) can be obtained their uniform-norm counterparts. This information is known for a long time, (see e.g., [20] for definition of HBS). This elegant method was generalized to some variable exponent Lebesgue spaces functions defined on R (see Theorem 1 of [9]).…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

Exponential approximation in variable exponent Lebesgue spaces on the real line

Akgün

2022

Constructive Mathematical Analysis

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Landau-Kolmogorov inequalities for semigroups and groups

Certain¹,

Kurtz²

1977

Proc. Amer. Math. Soc.

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Abstract.An elementary functional analytic argument is given showing how inequalities of the form \\f(k)\f < A^ll/H""*!!/00!!*, where/is a real, n-times differentiable function and ||-|| denotes the sup norm on (0, oo) (or (-00,00)), yield corresponding inequalities, \Akx\" < ^¿M"-\A"x\ , for generators of linear contraction semigroups (or groups) on arbitrary Banach spaces with norm ||. Since Landau, Kolmogorov, Schoenberg and Cavaretta have established the function inequalities with the best possible constants, this argument gives the generator inequalities with the best possible constants for general Banach spaces extending work of Kallman and Rota, Hille and others. Questions concerning the best possible constants for specific Banach spaces remain open.

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Inverse Theorems for Functions in L p and other Spaces

Cited by 2 publications

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Exponential approximation in variable exponent Lebesgue spaces on the real line

Exponential approximation in variable exponent Lebesgue spaces on the real line

Landau-Kolmogorov inequalities for semigroups and groups

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