Open Access
Translator Disclaimer
May/June 2013 Approximation in variation by homothetic operators in multidimensional setting
Laura Angeloni, Gianluca Vinti
Differential Integral Equations 26(5/6): 655-674 (May/June 2013). DOI: 10.57262/die/1363266083


This paper deals with approximation problems in a multidimensional setting for a family of convolution integral operators of homothetic type of the form $$ (T_wf)({s}) = \int_{\mathcal R^N} K_w ({ t}) f({ st}) \,d{ t},~~w>0,~ { s} \in \mathcal R^N; $$ here $\{K_w\}_{w>0}$ are approximate indentities, $\mathcal R:=\mathbb R^+_0$, and $f$ belongs to $BV^{\varphi}(\mathcal R^N)$, the space of functions with bounded $\varphi$-variation on $\mathcal R^N$. The main result of the paper establishes that, if $f$ is $\varphi$-absolutely continuous, there exists $\mu >0$ such that $$ V^{\varphi}[\mu(T_w f-f)]\longrightarrow 0,\ \ \hbox{as}\ w\rightarrow +\infty, $$ where $V^{\varphi}$ denotes the multidimensional $\varphi$-variation introduced in [8]. The rate of approximation is also investigated, by means of suitable Lipschitz classes.


Download Citation

Laura Angeloni. Gianluca Vinti. "Approximation in variation by homothetic operators in multidimensional setting." Differential Integral Equations 26 (5/6) 655 - 674, May/June 2013.


Published: May/June 2013
First available in Project Euclid: 14 March 2013

zbMATH: 1299.41026
MathSciNet: MR3086404
Digital Object Identifier: 10.57262/die/1363266083

Primary: 26A45 , 26B30 , 41A25 , 41A35 , 47G10

Rights: Copyright © 2013 Khayyam Publishing, Inc.


Vol.26 • No. 5/6 • May/June 2013
Back to Top