On some Sobolev spaces with matrix weights and classical type Sobolev orthogonal polynomials

Zagorodnyuk, Sergey M.

doi:10.1080/10236198.2021.1887160

Cited by 7 publications

(8 citation statements)

References 24 publications

(33 reference statements)

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“…In the general case, since pu0 ðÞ ðÞ ¼ p 0 ðÞ 6 ¼ 0, there also exists a suitable R 1 , which guarantees that F is analytic in C T where φ n t ðÞare some complex-valued functions. Then the function φ n t ðÞis a complex polynomial of degree n, ∀n ∈  þ , see [28,Lemma 3.5]. Suppose that degp ≥ 1, and…”

Section: Pencils Of Banded Matrices and Sobolev Orthogonalitymentioning

confidence: 99%

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Pencils of Semi-Infinite Matrices and Orthogonal Polynomials

Zagorodnyuk¹

2023

Matrix Theory - Classics and Advances

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Semi-infinite matrices, generalized eigenvalue problems, and orthogonal polynomials are closely related subjects. They connect different domains in mathematics—matrix theory, operator theory, analysis, differential equations, etc. The classical examples are Jacobi and Hessenberg matrices, which lead to orthogonal polynomials on the real line (OPRL) and orthogonal polynomials on the unit circle (OPUC). Recently there turned out that pencils (i.e., operator polynomials) of semi-infinite matrices are related to various orthogonal systems of functions. Our aim here is to survey this increasing subject. We are mostly interested in pencils of symmetric semi-infinite matrices. The corresponding polynomials are defined as generalized eigenvectors of the pencil. These polynomials possess special orthogonality relations. They have physical and mathematical applications that will be discussed. Examples show that there is an unclarified relation to Sobolev orthogonal polynomials. This intriguing connection is a challenge for further investigations.

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Section: Pencils Of Banded Matrices and Sobolev Orthogonalitymentioning

confidence: 99%

“…Corollary 1 ( [28]) In conditions of Theorem 1.4 suppose that g n t ðÞ¼t n , n ∈  þ ; fw ðÞ ¼ 1, uw ðÞ ¼ w. Polynomials φ n t ðÞ fg ∞ n¼0 satisfy the following recurrence relation:…”

Section: Pencils Of Banded Matrices and Sobolev Orthogonalitymentioning

confidence: 99%

Pencils of Semi-Infinite Matrices and Orthogonal Polynomials

Zagorodnyuk¹

2023

Matrix Theory - Classics and Advances

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“…By W 2 (M ) we denote the closure of A 2 (M ) in the norm of L 2 (M ). The subspace W 2 (M ) is said to be the Sobolev space with the matrix measure M , see [6]. Elements of A 2 (M ) will be also denoted by their first components.…”

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confidence: 99%

On the multiplication operator by an independent variable in matrix Sobolev spaces

Zagorodnyuk¹

2022

Preprint

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We study the operator A of multiplication by an independent variable in a matrix Sobolev space W 2 (M ). In the cases of finite measures on [a, b] with (2 × 2) and (3 × 3) real continuous matrix weights of full rank it is shown that the operator A is symmetrizable. Namely, there exist two symmetric operators B and C in a larger space such that Af = CB −1 f , f ∈ D(A). As a corollary, we obtain some new orthogonality conditions for the associated Sobolev orthogonal polynomials. These conditions involve two symmetric operators in an indefinite metric space.

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“…In general, polynomials P n (x) and L n (x) turns out to be Sobolev orthogonal polynomials on the real line with some explicit matrix measures. This can be derived on a way proposed in papers [16] and [17].…”

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confidence: 99%

“…However our manipulations are not formal, but we take care about the convergence questions. The case of L n is simpler, and we can check the required relation (17) by expanding the generating function into the Taylor series.…”

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confidence: 99%