BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Metric approximation of set-valued functions by integral operators - Elena Berdysheva (University of Cape Town) DTSTART:20240715T111500Z DTEND:20240715T113500Z UID:TALK218125@talks.cam.ac.uk DESCRIPTION:We study approximation by integral operators of set-valued fun ctions (SVFs) mapping a compact interval [a\,b] into the space of compact nonempty subsets of Rd.\n \;\nOlder works on approximation of SVFs con sider 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 approximat ion methods deliver approximants whose values are convex\, even if the fun ction to be approximated did not have this property. Clearly\, such method s are useless when one wants to approximate a set-valued function with gen eral\, not necessarily convex values.\n \;\nA pioneering work on appro ximation of SVFs with general values was done by Z. Artstein who construct ed piecewise linear approximantis based of special pairs of points that ar e termed in later works ``metric pairs''. Using the concept of metric pair s\, N. Dyn\, E. Farkhi and A. Mokhov developed in a series of works techni ques that are appropriate for work with SVFs with general\, not necessaril y convex values.\n \;\nIn this talk I will describe a construction tha t adapts integral approximation operators to set-valued functions with gen eral (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.\n \ ;\nJoint work with N. Dyn\, E. Farkhi and A. Mokhov. LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR