Inverse results of approximation and the saturation order for the sampling Kantorovich series
type: Item
instance of: scholarly article
title: Inverse results of approximation and the saturation order for the sampling Kantorovich series
zbMATH Open document ID: 1416.41019
author: Danilo Costarelli
author: Gianluca Vinti
published in: Journal of Approximation Theory
publication date: 2019-05-02T00:00:00Z
review text: 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 \[ \…
reviewed by: B. Szal
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 7050071
zbMATH Keywords: sampling Kantorovich series
zbMATH Keywords: saturation order
zbMATH Keywords: Voronovskaja formula
zbMATH Keywords: generalized sampling operator
zbMATH Keywords: inverse theorem
MaRDI profile type: Publication
full work available at URL: j.jat.2019.03.001
OpenAlex ID: W2920935281
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DOI: 10.1016/J.JAT.2019.03.001
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