Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation

Handle URI:
http://hdl.handle.net/10754/621379
Title:
Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation
Authors:
Ait-Haddou, Rachid; Mazure, Marie-Laurence
Abstract:
On a closed bounded interval, consider a nested sequence of Extended Chebyshev spaces possessing Bernstein bases. This situation automatically generates an infinite dimension elevation algorithm transforming control polygons of any given level into control polygons of the next level. The convergence of these infinite sequences of polygons towards the corresponding curves is a classical issue in computer-aided geometric design. Moreover, according to recent work proving the existence of Bernstein-type operators in such Extended Chebyshev spaces, this nested sequence is automatically associated with an infinite sequence of Bernstein operators which all reproduce the same two-dimensional space. Whether or not this sequence of operators converges towards the identity on the space of all continuous functions is a natural issue in approximation theory. In the present article, we prove that the two issues are actually equivalent. Not only is this result interesting on the theoretical side, but it also has practical implications. For instance, it provides us with a Korovkin-type theorem of convergence of any infinite dimension elevation algorithm. It also enables us to tackle the question of convergence of the dimension elevation algorithm for any nested sequence obtained by repeated integration of the kernel of a given linear differential operator with constant coefficients. © 2016 Springer Science+Business Media New York
KAUST Department:
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
Citation:
Ait-Haddou R, Mazure M-L (2016) Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation. Constructive Approximation 43: 425–461. Available: http://dx.doi.org/10.1007/s00365-016-9331-9.
Publisher:
Springer Science + Business Media
Journal:
Constructive Approximation
Issue Date:
18-Mar-2016
DOI:
10.1007/s00365-016-9331-9
Type:
Article
ISSN:
0176-4276; 1432-0940
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorAit-Haddou, Rachiden
dc.contributor.authorMazure, Marie-Laurenceen
dc.date.accessioned2016-11-03T08:27:55Z-
dc.date.available2016-11-03T08:27:55Z-
dc.date.issued2016-03-18en
dc.identifier.citationAit-Haddou R, Mazure M-L (2016) Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation. Constructive Approximation 43: 425–461. Available: http://dx.doi.org/10.1007/s00365-016-9331-9.en
dc.identifier.issn0176-4276en
dc.identifier.issn1432-0940en
dc.identifier.doi10.1007/s00365-016-9331-9en
dc.identifier.urihttp://hdl.handle.net/10754/621379-
dc.description.abstractOn a closed bounded interval, consider a nested sequence of Extended Chebyshev spaces possessing Bernstein bases. This situation automatically generates an infinite dimension elevation algorithm transforming control polygons of any given level into control polygons of the next level. The convergence of these infinite sequences of polygons towards the corresponding curves is a classical issue in computer-aided geometric design. Moreover, according to recent work proving the existence of Bernstein-type operators in such Extended Chebyshev spaces, this nested sequence is automatically associated with an infinite sequence of Bernstein operators which all reproduce the same two-dimensional space. Whether or not this sequence of operators converges towards the identity on the space of all continuous functions is a natural issue in approximation theory. In the present article, we prove that the two issues are actually equivalent. Not only is this result interesting on the theoretical side, but it also has practical implications. For instance, it provides us with a Korovkin-type theorem of convergence of any infinite dimension elevation algorithm. It also enables us to tackle the question of convergence of the dimension elevation algorithm for any nested sequence obtained by repeated integration of the kernel of a given linear differential operator with constant coefficients. © 2016 Springer Science+Business Media New Yorken
dc.publisherSpringer Science + Business Mediaen
dc.subjectApproximation by Bernstein-type operatorsen
dc.subjectBlossomsen
dc.subjectDimension elevationen
dc.subjectExtended Chebyshev spacesen
dc.subjectGeometric designen
dc.titleApproximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevationen
dc.typeArticleen
dc.contributor.departmentKing Abdullah University of Science and Technology, Thuwal, Saudi Arabiaen
dc.identifier.journalConstructive Approximationen
dc.contributor.institutionLaboratoire Jean Kuntzmann, CNRS, UMR 5224, Université Grenoble Alpes, BP 53, Grenoble Cedex 9, Franceen
kaust.authorAit-Haddou, Rachiden
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