## Differential and Integral Equations

- Differential Integral Equations
- Volume 26, Number 5/6 (2013), 655-674.

### Approximation in variation by homothetic operators in multidimensional setting

Laura Angeloni and Gianluca Vinti

#### Abstract

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.

#### Article information

**Source**

Differential Integral Equations, Volume 26, Number 5/6 (2013), 655-674.

**Dates**

First available in Project Euclid: 14 March 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1363266083

**Mathematical Reviews number (MathSciNet)**

MR3086404

**Zentralblatt MATH identifier**

1299.41026

**Subjects**

Primary: 26B30: Absolutely continuous functions, functions of bounded variation 26A45: Functions of bounded variation, generalizations 41A25: Rate of convergence, degree of approximation 41A35: Approximation by operators (in particular, by integral operators) 47G10: Integral operators [See also 45P05]

#### Citation

Angeloni, Laura; Vinti, Gianluca. Approximation in variation by homothetic operators in multidimensional setting. Differential Integral Equations 26 (2013), no. 5/6, 655--674. https://projecteuclid.org/euclid.die/1363266083