Metric approximation of set-valued functions by integral operators

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• Elena Berdysheva (University of Cape Town)
• Monday 15 July 2024, 12:15-12:35
• Seminar Room 1, Newton Institute.

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DREW01 - Multivariate approximation, discretization, and sampling recovery

We study approximation by integral operators of set-valued functions (SVFs) mapping a compact interval [a,b] into the space of compact nonempty subsets of Rd.   Older works on approximation of SVFs consider almost exclusively SVFs with convex values. The standard techniques used for work with SVFs were developed for convex sets and suffer from the phenomenon called convexification. As a result, corresponding approximation methods deliver approximants whose values are convex, even if the function to be approximated did not have this property. Clearly, such methods are useless when one wants to approximate a set-valued function with general, not necessarily convex values.   A pioneering work on approximation of SVFs with general values was done by Z. Artstein who constructed piecewise linear approximantis based of special pairs of points that are termed in later works ``metric pairs’’. Using the concept of metric pairs, N. Dyn, E. Farkhi and A. Mokhov developed in a series of works techniques that are appropriate for work with SVFs with general, not necessarily convex values.   In this talk I will describe a construction that adapts integral approximation operators to set-valued functions with general (not necessarily convex) compact images. The 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.   Joint work with N. Dyn, E. Farkhi and A. Mokhov.

This talk is part of the Isaac Newton Institute Seminar Series series.

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