# Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing

## TL;DR

This paper investigates the convergence in variation of multidimensional generalized sampling series with averaged kernels, introduces sampling-Kantorovich operators for L^p convergence, and applies these results to digital image smoothing techniques.

## Contribution

It establishes convergence in variation for multidimensional sampling series using new sampling-Kantorovich operators and links derivatives of these operators to the series, with applications in image processing.

## Key findings

- Proves convergence in L^p for a class of operators.
- Establishes convergence in variation for sampling series.
- Demonstrates applications to digital image smoothing.

## Abstract

In this paper we study the problem of the convergence in variation for the generalized sampling series based upon averaged-type kernels in the multidimensional setting. As a crucial tool, we introduce a family of operators of sampling-Kantorovich type for which we prove convergence in L^p on a subspace of L^p(R^N): therefore we obtain the convergence in variation for the multidimensional generalized sampling series by means of a relation between the partial derivatives of such operators acting on an absolutely continuous function f and the sampling-Kantorovich type operators acting on the partial derivatives of f. Applications to digital image processing are also furnished.