Convergence and rate of approximation in $BV^{\varphi}(\mathbb{R}^N_+)$ for a class of Mellin integral operators

TL;DR

This paper investigates the convergence and approximation rates of Mellin-type integral operators within the space of functions having bounded $ extit{ extvarphi}$-variation on the positive real space, expanding understanding of their approximation properties.

Contribution

It introduces new convergence and approximation rate results for Mellin integral operators acting on functions with bounded $ extit{ extvarphi}$-variation in multiple dimensions.

Findings

Established convergence results for Mellin operators in $BV^{ ext{ extvarphi}}( e^N_+)$.

Derived explicit rates of approximation for these operators.

Extended the theory of approximation in spaces of functions with bounded variation.

Abstract

In this paper we study convergence results and rate of approximation for a family of linear integral operators of Mellin type in the frame of $BV^{\varphi}(\mathbb{R}^N_+)$. Here $BV^{\varphi}(\mathbb{R}^N_+)$ denotes the space of functions with bounded $\varphi-$variation on $\mathbb{R}^N_+$, defined by means of a concept of multidimensional $\varphi-$variation in the sense of Tonelli.