Publication: Sequentially ordered Sobolev inner product and Laguerre-Sobolev polynomials
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2023-04-02
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MDPI
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Abstract
We study the sequence of polynomials {Sn}n≥0
that are orthogonal with respect to the general discrete Sobolev-type inner product ⟨f,g⟩s=∫f(x)g(x)dμ(x)+∑Nj=1∑djk=0λj,kf(k)(cj)g(k)(cj),
where μ
is a finite Borel measure whose support supp(μ)
is an infinite set of the real line, λj,k≥0
, and the mass points ci
, i=1,…,N
are real values outside the interior of the convex hull of supp(μ)
(ci∈R\Ch(supp(μ))∘)
. Under some restriction of order in the discrete part of ⟨⋅,⋅⟩s
, we prove that Sn
has at least n−d∗
zeros on Ch(supp(μ))∘
, being d∗
the number of terms in the discrete part of ⟨⋅,⋅⟩s
. Finally, we obtain the outer relative asymptotic for {Sn}
in the case that the measure μ
is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of ⟨⋅,⋅⟩s.
Description
This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications.
Keywords
Orthogonal polynomials, Sobolev orthogonality, Zeros location, Asymptotic behavior
Bibliographic citation
Díaz-González, A., Hernández, J., & Pijeira-Cabrera, H. (2023). Sequentially Ordered Sobolev Inner Product and Laguerre–Sobolev Polynomials. Mathematics, 11(8), 1956.