Metric Fourier approximation of set-valued functions of bounded variation

TL;DR

This paper develops a Fourier series-based approximation method for set-valued functions of bounded variation, providing convergence results and error bounds using new notions of continuity moduli.

Contribution

It introduces a novel Fourier approximation framework for set-valued functions, including error analysis and convergence properties in the Hausdorff metric.

Findings

Sequence of partial sums converges pointwise at continuity points.

Error bounds are established using new continuity moduli.

Convergence extends to certain sets at discontinuities.

Abstract

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.