Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points

Baier, Robert; Farkhi, Elza

doi:10.1007/s11228-006-0038-0

Cited by 42 publications

(14 citation statements)

References 30 publications

(27 reference statements)

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“…Corollary 5.8 implies the following "inclusion property" of the weighted metric integral as stated below. To show the right inclusion in (11), we use (10) and (8) and write…”

Section: F(x N ))mentioning

confidence: 99%

See 1 more Smart Citation

Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

Berdysheva

Dyn

Farkhi

et al. 2021

J Fourier Anal Appl

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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.

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“…Corollary 5.8 implies the following "inclusion property" of the weighted metric integral as stated below. To show the right inclusion in (11), we use (10) and (8) and write…”

Section: F(x N ))mentioning

confidence: 99%

“…Research on approximation and numerical integration of set-valued functions with convex images can be found e.g. in [6][7][8][9][10][11]14,[16][17][18]20,[27][28][29][30][31][32]36,37]. The standard tools used are the Minkowski linear combinations and the Aumann integral.…”

Section: Introductionmentioning

confidence: 99%

Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

Berdysheva

Dyn

Farkhi

et al. 2021

J Fourier Anal Appl

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“…Measures with finite support in S n−1 (class FM) are convex combination of measures in AM (cf. [4]). CM denotes either the family of measures AM or FM.…”

Section: Preliminariesmentioning

confidence: 99%

“…Definition 1.1 equals the definition given in [9] (cf. [4,Lemma 3.3]), where the norm-minimal element of Y (p, C) is used instead of the Steiner center. However, the definition above from [4] generalizes the GS-point from smooth measures to measures with finite support.…”

Section: Preliminariesmentioning

confidence: 99%

Generalized Steiner Selections Applied to Standard Problems of Set-Valued Numerical Analysis

Baier

2007

Progress in Nonlinear Differential Equations and Their Applications

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Generalized Steiner points and the corresponding selections for setvalued maps share interesting commutation properties with set operations which make them suitable for the set-valued numerical problems presented here. This short overview will present first applications of these selections to standard problems in this area, namely representation of convex, compact sets in R n and set operations, set-valued integration and interpolation as well as the calculation of attainable sets of linear differential inclusions. Hereby, the convergence results are given uniformly for a dense countable representation of generalized Steiner points/selections. To achieve this aim, stronger conditions on the set-valued map F have to be taken into account, e.g. the Lipschitz condition on F has to be satisfied for the Demyanov distance instead of the Hausdorff distance. To establish an overview on several applications, not the strongest available results are formulated in this article.

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“…Research on approximation and numerical integration of set-valued functions with convex images can be found e.g. in [37,16,36,29,30,32,31,17,9,10,27,7,18,20,8,28,11,6,14]. The standard tools used are the Minkowski linear combinations and the Aumann integral.…”

Section: Introductionmentioning

confidence: 99%