Metric approximation of set-valued functions of bounded variation

Berdysheva, Elena E.; Dyn, Nira; Farkhi, Elza; Mokhov, Andrey

doi:10.1016/j.cam.2018.09.039

Cited by 6 publications

(6 citation statements)

References 21 publications

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“…where a n ( f ) and b n ( f ) are the cosine and sine Fourier coefficients (13) of the function f . By Lemma 6.1 we have…”

Section: Example 74 Consider the Set-valued Functionmentioning

confidence: 99%

“…The tools used in these techniques include repeated binary metric averages [19,22,24], metric linear combinations [21,22], metric selections [22,23] and the metric integral [23], which is extended here to a weighted metric integral. In [13,[21][22][23] the authors studied approximation of set-valued functions by means of metric adaptations of classical approximation operators such as the Bernstein polynomial operator, the Schoenberg spline operator, the polynomial interpolation operator. While in older papers the approximated SVFs are mainly continuous, the later works [13,23] are concerned with multifunctions of bounded variation.…”

Section: Introductionmentioning

confidence: 99%

“…In [13,[21][22][23] the authors studied approximation of set-valued functions by means of metric adaptations of classical approximation operators such as the Bernstein polynomial operator, the Schoenberg spline operator, the polynomial interpolation operator. While in older papers the approximated SVFs are mainly continuous, the later works [13,23] are concerned with multifunctions of bounded variation.…”

Section: Introductionmentioning

confidence: 99%

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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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“…where a n ( f ) and b n ( f ) are the cosine and sine Fourier coefficients (13) of the function f . By Lemma 6.1 we have…”

Section: Example 74 Consider the Set-valued Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

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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“…For instance, in [2], binary metric average of sets is defined. In [13], Minkowski sum of two sets is used, and in [9] the notion of the sum specified in [2] has been extended, which is known as a metric linear combination of sets. In this paper, we have taken the Minkowski sum of sets.…”

Section: Introductionmentioning

confidence: 99%

“…Instead of Minkowski's sum of sets, he used the set of a sum of special pairs of elements, later known as "metric pairs". One may refer [9,14] for some more research on the approximation of compact set-valued maps. In this paper, we have studied the fractal approximation of set-valued maps.…”

Section: Introductionmentioning

confidence: 99%

Set-valued α-fractal functions

Pandey¹,

Som²,

Verma³

2022

Preprint

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In this paper, we introduce the concept of the α-fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its α-fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an α-fractal function is an attractor of an iterated function system.

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