TY - JOUR

T1 - Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

AU - Berdysheva, Elena E.

AU - Dyn, Nira

AU - Farkhi, Elza

AU - Mokhov, Alona

N1 - Publisher Copyright:
© 2021, The Author(s).

PY - 2021/4

Y1 - 2021/4

N2 - 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.

AB - 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.

KW - Compact sets

KW - Function of bounded variation

KW - Metric approximation operators

KW - Metric integral

KW - Metric linear combinations

KW - Metric selections

KW - Set-valued functions

KW - Trigonometric Fourier approximation

UR - http://www.scopus.com/inward/record.url?scp=85102364186&partnerID=8YFLogxK

U2 - 10.1007/s00041-021-09812-7

DO - 10.1007/s00041-021-09812-7

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AN - SCOPUS:85102364186

SN - 1069-5869

VL - 27

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

IS - 2

M1 - 17

ER -