Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

Elena E. Berdysheva, Nira Dyn, Elza Farkhi, Alona Mokhov

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number17
JournalJournal of Fourier Analysis and Applications
Volume27
Issue number2
DOIs
StatePublished - Apr 2021

Keywords

  • Compact sets
  • Function of bounded variation
  • Metric approximation operators
  • Metric integral
  • Metric linear combinations
  • Metric selections
  • Set-valued functions
  • Trigonometric Fourier approximation

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