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.
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. https://doi.org/10.57262/die/1363266083
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