For set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space... more For set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space of compact non-empty subsets of R d , we study approximation based on the metric approach that includes metric linear combinations, metric selections and weighted metric integrals. In our earlier papers we considered convergence of metric Fourier approximations and metric adaptations of some classical integral approximating operators for SVFs of bounded variation with compact graphs. While the pointwise limit of a sequence of these approximants at a point of continuity x of the set-valued function F is F (x), the limit set at a jump point was earlier described in terms of the metric selections of the multifunction. Here we show that, under certain assumptions on F , the limit set at x equals the metric average of the left and the right limits of F at x, thus extending the case of real-valued functions.
Journal of Fourier Analysis and Applications, 2021
We introduce and investigate an adaptation of Fourier series to set-valued functions (multifuncti... more 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.
Journal of Computational and Applied Mathematics, 2019
In this paper we approximate univariate set-valued functions (SVFs) of bounded variation with ran... more In this paper we approximate univariate set-valued functions (SVFs) of bounded variation with range consisting of general (not necessarily convex) compact sets. The approximation operators adapted to SVFs are local operators such as the symmetric Schoenberg spline operator, the Bernstein polynomial operator and the Steklov function. All operators are adapted by using metric linear combinations. Error bounds, obtained in the averaged Hausdorff metric, provide rates of approximation similar to those for real-valued functions of bounded variation.
This paper introduces a new integral of univariate set-valued functions of bounded variation with... more This paper introduces a new integral of univariate set-valued functions of bounded variation with compact images in R d. The new integral, termed the metric integral, is defined using metric linear combinations of sets and is shown to consist of integrals of all the metric selections of the integrated multifunction. The metric integral is a subset of the Aumann integral, but in contrast to the latter, it is not necessarily convex. For a special class of segment functions equality of the two integrals is shown. Properties of the metric selections and related properties of the metric integral are studied. Several indicative examples are presented.
This paper investigates the approximation of set-valued functions with compact images (not necess... more This paper investigates the approximation of set-valued functions with compact images (not necessarily convex), by adaptations of the Schoenberg spline operators and the Bernstein polynomial operators. When replacing the sum between numbers in these operators, by the Minkowski sum between sets, the resulting operators approximate only set valued functions with compact-convex images [10]. To obtain operators which approximate set-valued functions with compact images, we use the well known fact that both types of operators for realvalued functions can be evaluated by repeated binary weighted averages, starting from pairs of function values. Replacing the binary weighted averages between numbers by a binary operation between compact sets, introduced in [1] and termed in [4] the "metric average", we obtain operators which are defined for set-valued functions. We prove that the Schoenberg operators so defined approximate setvalued functions which are Hölder continuous, while for the Bernstein operators we prove approximation only for Lipschitz continuous set-valued functions with images in R all of the same topology. Examples illustrating the approximation results are presented.
Download this significant ebook and read on the Approximation Of Set Valued Functions Adaptation ... more Download this significant ebook and read on the Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators Ebook ebook. You won't find this ebook anywhere online. See the any books and unless you have lots of time to learn, it is possible to download some ebooks to your device and check. Are you currently hunt Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators? You then return to the right place to get the Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators Ebook. Read any ebook on line with easy actions. But if you would like to get it you can download a lot of ebooks. In looking over this particular guide, one to keep in your mind is that never fear and never be amazed to learn. Also you won't be given true idea by helpful information, it's very likely to create dream. Yes, attainable obtaining the future that is fantastic. But, it's not only type of imagination. Here's the time for you really to create ideal suggestions to create improved future. How exactly is by getting Process on Website Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators eBook among the analyzing material. You may possibly well be therefore treated to see it because it gives advantages and more opportunities for life. Though famous, to conclude this sort of ebook, then you possibly will not need to get it at once within a day. Doing the actions could enable you to feel bored. If you try to make looking at, possibly you'll approach other pursuits that are compelling. Nevertheless one of fundamentals we'd like one to get this kind of ebook will probably be that it'll not necessarily allow one to feel bored. Bored whenever is going to be merely in the event you never such as book. 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The paper is an updated survey of our work on the approximation of univariate set- valued functio... more The paper is an updated survey of our work on the approximation of univariate set- valued functions by samples-based linear approximation operators, beyond the results reported in our previous overview. Our approach is to adapt operators for real-valued functions to set-valued functions, by replacing operations between numbers by oper- ations between sets. For set-valued functions with compact convex images we use Minkowski convex combinations of sets, while for those with general compact images metric averages and metric linear combinations of sets are used. We obtain general approximation results and apply them to Bernstein polynomial operators, Schoenberg spline operators and polynomial interpolation operators.
In this work univariate set-valued functions (SVFs, multifunctions) with 1D compact sets as image... more In this work univariate set-valued functions (SVFs, multifunctions) with 1D compact sets as images are considered. For such a continuous SFV of bounded variation (CBV multifunction), we show that the boundaries of its graph are continuous, and inherit the continuity properties of the SVF. Based on these results we introduce a special class of representations of CBV multifunctions with a finite number of 'holes' in their graphs. Each such representation is a finite union of SVFs with compact convex images having boundaries with continuity properties as those of the represented SVF. With the help of these representations, positive linear operators are adapted to SVFs. For specific positive approximation operators error estimates are obtained in terms of the continuity properties of the approximated multifunction.
In this work, we introduce new approximation operators for univariate setvalued functions with ge... more In this work, we introduce new approximation operators for univariate setvalued functions with general compact images. We adapt linear approximation methods for real-valued functions by replacing linear combinations of numbers with new metric linear combinations of finite sequences of compact sets, thus obtaining "metric analogues" operators for set-valued functions. The new metric linear combination extends the binary metric average of Artstein. Approximation estimates for the metric analogue operators are derived. As examples we study metric Bernstein operators, metric Shoenberg operators and metric polynomial interpolants.
For set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space... more For set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space of compact non-empty subsets of R d , we study approximation based on the metric approach that includes metric linear combinations, metric selections and weighted metric integrals. In our earlier papers we considered convergence of metric Fourier approximations and metric adaptations of some classical integral approximating operators for SVFs of bounded variation with compact graphs. While the pointwise limit of a sequence of these approximants at a point of continuity x of the set-valued function F is F (x), the limit set at a jump point was earlier described in terms of the metric selections of the multifunction. Here we show that, under certain assumptions on F , the limit set at x equals the metric average of the left and the right limits of F at x, thus extending the case of real-valued functions.
Journal of Fourier Analysis and Applications, 2021
We introduce and investigate an adaptation of Fourier series to set-valued functions (multifuncti... more 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.
Journal of Computational and Applied Mathematics, 2019
In this paper we approximate univariate set-valued functions (SVFs) of bounded variation with ran... more In this paper we approximate univariate set-valued functions (SVFs) of bounded variation with range consisting of general (not necessarily convex) compact sets. The approximation operators adapted to SVFs are local operators such as the symmetric Schoenberg spline operator, the Bernstein polynomial operator and the Steklov function. All operators are adapted by using metric linear combinations. Error bounds, obtained in the averaged Hausdorff metric, provide rates of approximation similar to those for real-valued functions of bounded variation.
This paper introduces a new integral of univariate set-valued functions of bounded variation with... more This paper introduces a new integral of univariate set-valued functions of bounded variation with compact images in R d. The new integral, termed the metric integral, is defined using metric linear combinations of sets and is shown to consist of integrals of all the metric selections of the integrated multifunction. The metric integral is a subset of the Aumann integral, but in contrast to the latter, it is not necessarily convex. For a special class of segment functions equality of the two integrals is shown. Properties of the metric selections and related properties of the metric integral are studied. Several indicative examples are presented.
This paper investigates the approximation of set-valued functions with compact images (not necess... more This paper investigates the approximation of set-valued functions with compact images (not necessarily convex), by adaptations of the Schoenberg spline operators and the Bernstein polynomial operators. When replacing the sum between numbers in these operators, by the Minkowski sum between sets, the resulting operators approximate only set valued functions with compact-convex images [10]. To obtain operators which approximate set-valued functions with compact images, we use the well known fact that both types of operators for realvalued functions can be evaluated by repeated binary weighted averages, starting from pairs of function values. Replacing the binary weighted averages between numbers by a binary operation between compact sets, introduced in [1] and termed in [4] the "metric average", we obtain operators which are defined for set-valued functions. We prove that the Schoenberg operators so defined approximate setvalued functions which are Hölder continuous, while for the Bernstein operators we prove approximation only for Lipschitz continuous set-valued functions with images in R all of the same topology. Examples illustrating the approximation results are presented.
Download this significant ebook and read on the Approximation Of Set Valued Functions Adaptation ... more Download this significant ebook and read on the Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators Ebook ebook. You won't find this ebook anywhere online. See the any books and unless you have lots of time to learn, it is possible to download some ebooks to your device and check. Are you currently hunt Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators? You then return to the right place to get the Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators Ebook. Read any ebook on line with easy actions. But if you would like to get it you can download a lot of ebooks. In looking over this particular guide, one to keep in your mind is that never fear and never be amazed to learn. Also you won't be given true idea by helpful information, it's very likely to create dream. Yes, attainable obtaining the future that is fantastic. But, it's not only type of imagination. Here's the time for you really to create ideal suggestions to create improved future. How exactly is by getting Process on Website Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators eBook among the analyzing material. You may possibly well be therefore treated to see it because it gives advantages and more opportunities for life. Though famous, to conclude this sort of ebook, then you possibly will not need to get it at once within a day. Doing the actions could enable you to feel bored. If you try to make looking at, possibly you'll approach other pursuits that are compelling. Nevertheless one of fundamentals we'd like one to get this kind of ebook will probably be that it'll not necessarily allow one to feel bored. Bored whenever is going to be merely in the event you never such as book. Process on Website Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators txt Ebook delivers exactly what exactly every one wants. Produce no mistake, this particular guide is truly suggested for you. Your fascination relating to this Process on Website Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators AZW is going to be resolved sooner when just starting to learn. Moreover, whenever you finish this guide, may not only resolve your curiosity but additionally locate the significance that is genuine. Each expression contains a really excellent significance and also the selection of word is remarkable. The author with this guide is an great person. Free Download Books Download Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators LRX Everyone knows that reading Get Free Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators LRF is beneficial, because we will get advice on the web. Technology is now evolved, and reading Nibs College Ebook novels might be easier and much easier. We are able to see novels on the cellphone, pills and Kindle, etc. Thus, there are books getting into PDF format. At which one can acquire as much knowledge as you would like for downloading free of charge PDF books, Below sites. It may be brought by you based on the Process on Website Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators MS Word weblink on this report In case Get Free Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators EPUB you imagine difficult to acquire this kind of ebook. This is not only how you obtain the novel Process on Website Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators LRF to read. It's all about the consideration that one may acquire whenever. [PDF] as a way to achieve it is far from provided with this particular site. There are Process on Website Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators Fb2 the most current ebook to learn During clicking on the bond. Really, here it is! Download Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators DJVU E book goes along with this fresh advice as well as theory anytime anybody Using Available Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators IBA reading the information for this particular e book, sometimes few, you get exactly why is you feel fulfilled. That demonstration through reading it can be therefore streamlined, nonetheless have an impact on connected with the could be amazing this is. Nibs College Ebook Everybody could require that further periods to assist you realize more concerning this book. For people with accomplished content and articles connected with Get Free Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators PDF [PDF], then it's easy to really see the way great need of a book, regardless of the e book is undoubtedly,If you are interested in this sort of guide Download Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators Fb2, only make it immediately after potential. Information that is addiitional can be shown by Every one for people. You may obtain innovative items to attend in your everyday activity. Should they be practically all poured, anyone may make cutting-edge ecosystem. This offers some locations of the Download Approximation Of Set Valued Functions Adaptation Of Classical Approximation Operators IBA [PDF] that you might take. And when anybody really require a novel to relish a publication, decide the following guide nearly as superior reference.Some individuals may very well be joking when watching anybody reading within your spare time. Some might be shown admiration for connected with you personally. As well as a few might wish end anybody up with reading hobby. Why don't you believe carefully your presume? You have thought best? Looking at is a hobby along with a prerequisite during once. Be approximation-of-set-valued-functions-adaptation-of-classical-approximation-operators.pdf
The paper is an updated survey of our work on the approximation of univariate set- valued functio... more The paper is an updated survey of our work on the approximation of univariate set- valued functions by samples-based linear approximation operators, beyond the results reported in our previous overview. Our approach is to adapt operators for real-valued functions to set-valued functions, by replacing operations between numbers by oper- ations between sets. For set-valued functions with compact convex images we use Minkowski convex combinations of sets, while for those with general compact images metric averages and metric linear combinations of sets are used. We obtain general approximation results and apply them to Bernstein polynomial operators, Schoenberg spline operators and polynomial interpolation operators.
In this work univariate set-valued functions (SVFs, multifunctions) with 1D compact sets as image... more In this work univariate set-valued functions (SVFs, multifunctions) with 1D compact sets as images are considered. For such a continuous SFV of bounded variation (CBV multifunction), we show that the boundaries of its graph are continuous, and inherit the continuity properties of the SVF. Based on these results we introduce a special class of representations of CBV multifunctions with a finite number of 'holes' in their graphs. Each such representation is a finite union of SVFs with compact convex images having boundaries with continuity properties as those of the represented SVF. With the help of these representations, positive linear operators are adapted to SVFs. For specific positive approximation operators error estimates are obtained in terms of the continuity properties of the approximated multifunction.
In this work, we introduce new approximation operators for univariate setvalued functions with ge... more In this work, we introduce new approximation operators for univariate setvalued functions with general compact images. We adapt linear approximation methods for real-valued functions by replacing linear combinations of numbers with new metric linear combinations of finite sequences of compact sets, thus obtaining "metric analogues" operators for set-valued functions. The new metric linear combination extends the binary metric average of Artstein. Approximation estimates for the metric analogue operators are derived. As examples we study metric Bernstein operators, metric Shoenberg operators and metric polynomial interpolants.
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