Rational approximation and Sobolev-type orthogonality

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.

Rational approximation and Sobolev-type orthogonality Articles