Approximations of set-valued functions by metric linear operators

Nira Dyn*, Elza Farkhi, Alona Mokhov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

In this paper we introduce new approximation operators for univariate set-valued functions with general compact images in Rn. We adapt linear approximation methods for real-valued functions by replacing linear combinations of numbers with new metric linear combinations of finite sequences of compact sets, thus obtaining "metric analogues" of these operators for set-valued functions. The new metric linear combination extends the binary metric average of Artstein to several sets and admits any real coefficients. Approximation estimates for the metric analogue operators are derived. As examples we study metric Bernstein operators, metric Schoenberg operators, and metric polynomial interpolants.

Original languageEnglish
Pages (from-to)193-209
Number of pages17
JournalConstructive Approximation
Volume25
Issue number2
DOIs
StatePublished - Mar 2007

Keywords

  • Bernstein polynomial approximation
  • Compact sets
  • Linear approximation operators
  • Metric average
  • Minkowski linear combination
  • Piecewise linear set-valued functions
  • Polynomial interpolation
  • Schoenberg spline approximation
  • Selections
  • Set-valued functions

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