Sequentially Ordered Sobolev Inner Product and Laguerre–Sobolev Polynomials

Abel Díaz-González, Juan Hernández and Héctor Pijeira-Cabrera ()
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Abel Díaz-González: Department of Computer Science, Vanderbilt University, Nashville, TN 37240, USA
Juan Hernández: Escuela de Matemáticas, Universidad Autónoma de Santo Domingo, Santo Domingo 10105, Dominican Republic
Héctor Pijeira-Cabrera: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain

Mathematics, 2023, vol. 11, issue 8, 1-15

Abstract: We study the sequence of polynomials { S n } n ≥ 0 that are orthogonal with respect to the general discrete Sobolev-type inner product ⟨ f , g ⟩ s = ∫ f ( x ) g ( x ) d μ ( x ) + ∑ j = 1 N ∑ k = 0 d j λ j , k f ( k ) ( c j ) g ( k ) ( c j ) , where μ is a finite Borel measure whose support supp μ is an infinite set of the real line, λ j , k ≥ 0 , and the mass points c i , i = 1 , … , N are real values outside the interior of the convex hull of supp μ ( c i ∈ R \ C h ( supp ( μ ) ) ∘ ) . Under some restriction of order in the discrete part of ⟨ · , · ⟩ s , we prove that S n has at least n − d * zeros on C h ( supp μ ) ∘ , being d * the number of terms in the discrete part of ⟨ · , · ⟩ s . Finally, we obtain the outer relative asymptotic for { S n } 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 .

Keywords: orthogonal polynomials; Sobolev orthogonality; zeros location; asymptotic behavior (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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