Publication: Rational approximation and Sobolev-type orthogonality
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Publication date
2020-12
Defense date
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Tutors
Journal Title
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Publisher
Elsevier
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.