Rational approximation and Sobolev-type orthogonality Articles uri icon

publication date

  • December 2020

start page

  • 1

end page

  • 19

issue

  • 105481

volume

  • 260

International Standard Serial Number (ISSN)

  • 0021-9045

Electronic International Standard Serial Number (EISSN)

  • 1096-0430

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.

keywords

  • rational approximation; sobolev orthogonality; markov's theorem; zero location