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

Rachid Ait-Haddou

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)6-17
Number of pages12
JournalJournal of Approximation Theory
Volume181
DOIs
StatePublished - May 2014

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Mathematics(all)
  • Numerical Analysis

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