Metric Fourier approximation of set-valued functions of bounded\n variation

2020/08/24 by Elena E. Berdysheva, Berdysheva, Elena E., Nira Dyn +5
Mathematics · #26E25 #28B20 #28C20 #42A20 #42A99 #54C60 #54C65 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2008.10340

openalex publication_date 2020/08/24 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We introduce and investigate an adaptation of Fourier series to set-valued\nfunctions (multifunctions, SVFs) of bounded variation. In our approach we\ndefine an analogue of the partial sums of the Fourier series with the help of\nthe Dirichlet kernel using the newly defined weighted metric integral. We\nderive error bounds for these approximants. As a consequence, we prove that the\nsequence of the partial sums converges pointwisely in the Hausdorff metric to\nthe values of the approximated set-valued function at its points of continuity,\nor to a certain set described in terms of the metric selections of the\napproximated multifunction at a point of discontinuity. Our error bounds are\nobtained with the help of the new notions of one-sided local moduli and\nquasi-moduli of continuity which we discuss more generally for functions with\nvalues in metric spaces.\n

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