Publication:
Rational approximation and Sobolev-type orthogonality

Loading...
Thumbnail Image
Identifiers
Publication date
2020-12
Defense date
Advisors
Tutors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Impact
Google Scholar
Export
Research Projects
Organizational Units
Journal Issue
Abstract
In this paper, we study the sequence of orthogonal polynomials {Sn}∞ n=0 with respect to the Sobolev-type inner product ⟨ f, g⟩ = ∫ 1 −1 f (x)g(x) dµ(x) + ∑ N j=1 η j f (d j) (c j )g (d j) (c j ) where µ is a finite positive Borel measure whose support supp (µ) ⊂ [−1, 1] contains an infinite set of points, η j > 0, N, d j ∈ Z+ and {c1, . . . , cN } ⊂ R \ [−1, 1]. Under some restriction of order in the discrete part of ⟨·, ·⟩, we prove that for sufficiently large n the zeros of Sn are real, simple, n − N of them lie on (−1, 1) and each of the mass points c j “attracts” one of the remaining N zeros. The sequences of associated polynomials {S [k] n }∞ n=0 are defined for each k ∈ Z+. If µ is in the Nevai class M(0, 1), we prove an analogue of Markov’s Theorem on rational approximation to Markov type functions and prove that convergence takes place with geometric speed.
Description
Keywords
Rational approximation, Sobolev orthogonality, Markov's theorem, Zero location
Bibliographic citation
Díaz-González, A., Pijeira-Cabrera, H. & Pérez-Yzquierdo, I. (2020). Rational approximation and Sobolev-type orthogonality. Journal of Approximation Theory, 260, 105481.