Convergence in variation and a characterization of the absolute continuity

Abstract

We study approximation results for a family of Mellin integral operators of the form $\left(T_{w}f\right)\left(\mathtt{s}\right)={\int }_{{\mathbb{R}}_{+}^N}{K}_w(\mathtt{t},f(\mathtt{s}\mathtt{t}\left)\right)\frac{\mathrm{d}\mathtt{t}}{⟨\mathtt{t}⟩},\mathtt{s}\in {\mathbb{R}}_{+}^N,w>0,$ where $\{K_w{\}}_{w>0}$ is a family of kernels, $⟨\mathtt{t}⟩:=\prod _{i=1}^N{t}_i$, $\mathtt{t}=(t_1,\dots ,t_N)\in {\mathbb{R}}_{+}^N$, and f is a function of bounded variation on ${\mathbb{R}}_{+}^N$. The starting point of this study is motivated by the important applications that approximation properties of certain families of integral operators have in image reconstruction and in other fields. In order to treat such problems, to work in BV -spaces in the multidimensional setting of ${\mathbb{R}}_{+}^N$ becomes crucial: for this reason we use a multidimensional concept of variation in the sense of Tonelli, adapted from the classical definition to the present setting of ${\mathbb{R}}_{+}^N$ equipped with the Haar measure. Using such definition of variation, we obtain a convergence result proving that $V[T_{w}f-f]\to 0$, as $w\to +\mathrm{\infty }$, whenever f is an absolutely continuous function; moreover we also study the problem of the rate of approximation. In case of regular kernels, we finally prove a characterization of the absolute continuity in terms of the convergence in variation by means of the Mellin-type operators $\{T_{w}f{\}}_{w>0}$.

Keywords:

• Mellin integral operators
• convergence in variation
• absolutely continuous functions
• multidimensional variation

AMS Subject Classification:

• 41A35
• 41A25
• 26B30
• 26B99
• 26D10

Acknowledgments

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

The first author was supported within the GNAMPA-INdAM Project 2015 ‘Metodi di approssimazione e applicazioni al Signal e Image Processing’.

ORCID

L. Angeloni http://orcid.org/0000-0002-2214-6751

G. Vinti http://orcid.org/0000-0002-9875-2790