Inverse results of approximation and the saturation order for the sampling Kantorovich series
In this paper, the authors study the saturation order for the sampling Kantorovich series in the space of uniformly continuous and bounded functions. Let $K_{w}^{\chi}f$ for $w>0$ be the family of the sampling Kantorovich operators given by the following formula: \[ \left( K_{w}^{\chi}f\right) \left( x\right) :=\sum_{k\in \mathbb{Z}}\left[ w\int_{k/w}^{\left( k+1\right) /w}f\left( u\right) du\right] \chi \left( wx-k\right) \] for $x\in\mathbb{R}$, where $f:\mathbb{R} \rightarrow\mathbb{R}$ is a locally integrable functions for which the operators are well-defined and $\chi:\mathbb{R} \rightarrow\mathbb{R}$ denotes a suitable kernel function, which satisfies the usual conditions ([\textit{P. L. Butzer} and \textit{R. J. Nessel}, Fourier analysis and approximation. Vol. 1: One-dimensional theory. Birkhäuser Verlag, Basel (1971; Zbl 0217.42603)]). Moreover, let \[ \left( G_{w}^{\chi}f\right) \left( x\right) =\sum_{k\in \mathbb{Z}}f\left( \frac{k}{w}\right) \chi \left( wx-k\right) \] for $x\in\mathbb{R}$, where $f:\mathbb{R} \rightarrow\mathbb{R}$ is any function such that the above sampling Kantorovich series convergent for every $x\in\mathbb{R}$ and $\chi:\mathbb{R} \rightarrow \mathbb{R}$ is a kernel as above. Denote by $C_{B}^{U}\left(\mathbb{R}\right)$ the space of all uniformly continuous and bounded function $f:\mathbb{R} \rightarrow\mathbb{R}$. The authors proved the following theorems: Theorem 1. Let $f\in C_{B}^{U}\left(\mathbb{R}\right) $. Then \[ \left( K_{w}^{\chi}f\right) \left( x\right) =\left( G_{w}^{\overline{\chi}}F^{\prime}\right) \left( x+\frac{1}{2w}\right) \] for every $x\in\mathbb{R}$ and $w>0$, where $F\left( x\right) :=\int_{0}^{x}f\left( t\right) dt $ $\left( x\in\mathbb{R}\right) $ and $\overline{\chi}\left( t\right) =\int_{-1/2}^{1/2}\chi \left( t+v\right) dv$ $\left( t\in\mathbb{R}\right) $. Theorem 2. Let $f:\mathbb{R} \rightarrow\mathbb{R}$ be a differentiable function, such that $f^{\prime}\in C_{B}^{U}\left(\mathbb{R}\right) $. Then \[ \lim_{w\rightarrow \infty}w\left[ \left( G_{w}^{\chi}f\right) \left( x\right) -f\left( x\right) \right] =A_{1}^{\chi}f^{\prime}\left( x\right) \] uniformly with respect to $x\in\mathbb{R}$, where $A_{1}^{\chi}:=\sum_{k\in \mathbb{Z}}\left( k-u\right) \chi \left( u-k\right) $ for every $u\in\mathbb{R}$. Theorem 3. Let $\chi $ be a kernel for which $A_{1}^{\chi}\neq -\frac{1}{2}$ and let $ f\in C_{B}^{U}\left(\mathbb{R}\right) $ be such that \[ \left\Vert K_{w}^{\chi}f-f\right\Vert _{\infty}=o\left( w^{-1}\right) \] as $w\rightarrow \infty$. Then $f$ is constant over $\mathbb{R}$. Theorem 4. Let be a kernel satisfying the conditions: \begin{itemize} \item[1.] $\chi \in C^{1}\left(\mathbb{R}\right)$ and $M_{1}\left( \chi^{\prime}\right) :=\sup_u \sum_{k\in \mathbb{Z}}\left\vert k-u\right\vert \left\vert \chi^{\prime}\left( u-k\right) \right\vert <\infty$, \item[2.] the series $\sum_{k\in \mathbb{Z}}\left\vert \chi^{\prime}\left( wx-k\right) \right\vert $, $x\in \mathbb{R}$, $w>0$, is uniformly convergent on the compact subsets of $ \mathbb{R}$, with respect to the variable $x$. \end{itemize} Moreover, let $f\in C_{B}^{U}\left(\mathbb{R}\right)$ be such that \[ \left\Vert K_{w}^{\chi}f-f\right\Vert _{\infty}=O\left( w^{-\alpha}\right) \] as $w\rightarrow \infty $ with $\alpha \in \left( 0,1\right)$. Then $f$ belongs to $\operatorname{Lip}( \alpha)$.
scientific article; zbMATH DE number 7050071
scientific article