Dimension elevation in Müntz spaces: A new emergence of the Müntz condition

Handle URI:
http://hdl.handle.net/10754/563514
Title:
Dimension elevation in Müntz spaces: A new emergence of the Müntz condition
Authors:
Ait-Haddou, Rachid
Abstract:
We show that the limiting polygon generated by the dimension elevation algorithm with respect to the Müntz space span(1,tr1,tr2,trm,. . .), with 0 < r1 < r2 < ⋯ < r m < ⋯ and lim n →∞r n = ∞, over an interval [a, b] ⊂ ] 0, ∞ [ converges to the underlying Chebyshev-Bézier curve if and only if the Müntz condition ∑i=1∞1ri=∞ is satisfied. The surprising emergence of the Müntz condition in the problem raises the question of a possible connection between the density questions of nested Chebyshev spaces and the convergence of the corresponding dimension elevation algorithms. The question of convergence with no condition of monotonicity or positivity on the pairwise distinct real numbers r i remains an open problem. © 2014 Elsevier Inc.
KAUST Department:
Visual Computing Center (VCC)
Publisher:
Elsevier BV
Journal:
Journal of Approximation Theory
Issue Date:
May-2014
DOI:
10.1016/j.jat.2014.01.006
ARXIV:
arXiv:1309.0938
Type:
Article
ISSN:
00219045
Additional Links:
http://arxiv.org/abs/arXiv:1309.0938v1
Appears in Collections:
Articles; Visual Computing Center (VCC)

Full metadata record

DC FieldValue Language
dc.contributor.authorAit-Haddou, Rachiden
dc.date.accessioned2015-08-03T11:53:23Zen
dc.date.available2015-08-03T11:53:23Zen
dc.date.issued2014-05en
dc.identifier.issn00219045en
dc.identifier.doi10.1016/j.jat.2014.01.006en
dc.identifier.urihttp://hdl.handle.net/10754/563514en
dc.description.abstractWe show that the limiting polygon generated by the dimension elevation algorithm with respect to the Müntz space span(1,tr1,tr2,trm,. . .), with 0 < r1 < r2 < ⋯ < r m < ⋯ and lim n →∞r n = ∞, over an interval [a, b] ⊂ ] 0, ∞ [ converges to the underlying Chebyshev-Bézier curve if and only if the Müntz condition ∑i=1∞1ri=∞ is satisfied. The surprising emergence of the Müntz condition in the problem raises the question of a possible connection between the density questions of nested Chebyshev spaces and the convergence of the corresponding dimension elevation algorithms. The question of convergence with no condition of monotonicity or positivity on the pairwise distinct real numbers r i remains an open problem. © 2014 Elsevier Inc.en
dc.publisherElsevier BVen
dc.relation.urlhttp://arxiv.org/abs/arXiv:1309.0938v1en
dc.subjectChebyshev blossomingen
dc.subjectChebyshev-Bézier curvesen
dc.subjectChebyshev-Bernstein basesen
dc.subjectDimension elevationen
dc.subjectGelfond-Bernstein basesen
dc.subjectMüntz spacesen
dc.subjectSchur functionsen
dc.titleDimension elevation in Müntz spaces: A new emergence of the Müntz conditionen
dc.typeArticleen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.identifier.journalJournal of Approximation Theoryen
dc.identifier.arxividarXiv:1309.0938en
kaust.authorAit-Haddou, Rachiden
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