Metric approximation of set-valued functions of bounded variation by integral operators

TL;DR

This paper extends integral approximation operators to set-valued functions of bounded variation, providing pointwise and global error estimates, convergence results, and illustrating with Bernstein-Durrmeyer and Kantorovich operators.

Contribution

It introduces a novel adaptation of integral operators for set-valued functions using the weighted metric integral, with new error bounds and convergence analysis.

Findings

Pointwise convergence at points of continuity.

Error estimates at points of discontinuity.

Global error bounds using Hausdorff distance.

Abstract

We introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval $[a,b]$ into the space of compact non-empty subsets of ${\mathbb R}^d$. All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a set-valued function $F$, we obtain pointwise error estimates for sequences of integral operators at points of continuity, leading to convergence at such points to $F$. At points of discontinuity of $F$, we derive estimates, which yield the convergence to a set, first described in our previous work on the metric Fourier operator. Our analysis uses recently defined one-sided local quasi-moduli at points of discontinuity and several notions of local Lipschitz property at points of continuity. We also provide a global approach for error bounds. A multifunction $F$ is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in $L^1$ provides our global estimates. The theory is illustrated by presenting the examples of two concrete operators: the Bernstein-Durrmeyer operator and the Kantorovich operator.