THE POINTWISE LIMIT SET OF METRIC INTEGRAL OPERATORS APPROXIMATING SET-VALUED FUNCTIONS

E. E. Berdysheva, N. Dyn, E. Farkhi, A. Mokhov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

For set-valued functions (SVFs, multifunctions), mapping a compact interval [a,b] into the space of compact non-empty subsets of Rd, we study the pointwise limits of metric integral approximation operators. In our earlier papers, we have considered convergence of metric Fourier approximations and metric adaptations of some classical integral approximating operators for SVFs of bounded variation with compact graphs. While the pointwise limit of a sequence of these approximants at a point of continuity x of the set-valued function F is F(x), the limit set at a jump point is described in terms of the metric selections of the multifunction. In this paper, we show that, under certain assumptions on F, the limit set at x equals the metric average of the left and the right limits of F at x. This result extends the known classical theorems from the case of real-valued functions to SVFs.

Original languageEnglish
Pages (from-to)23-38
Number of pages16
JournalApplied Set-Valued Analysis and Optimization
Volume7
Issue number1
DOIs
StatePublished - 1 Apr 2025

Keywords

  • Functions of bounded variation
  • Integral operators
  • Metric approximation
  • metric Fourier approximation
  • Metric integral
  • Positive linear operators
  • Set-valued functions

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