Vector Interpretation of the Matrix Orthogonality on the Real Line

In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients...

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Bibliographic Details
Published inActa applicandae mathematicae Vol. 112; no. 3; pp. 357 - 383
Main Authors Branquinho, A., Marcellán, F., Mendes, A.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.12.2010
Springer Nature B.V
Subjects
Applications of Mathematics
Asymptotic methods
Calculus of Variations and Optimal Control; Optimization
Computational Mathematics and Numerical Analysis
Mathematics
Mathematics and Statistics
Matrix
Partial Differential Equations
Polynomials
Probability Theory and Stochastic Processes
Studies
Asymptotic results
33C45
39B42
Matrix orthogonal polynomials
Recurrence relation
Problems of Hermite-Padé
Tridiagonal operator
Nevai class
Favard theorem
Linear functional
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Summary:In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients that do not obey to any type of symmetry. In this sense the vectorial reinterpretation allows us to study a non-symmetric case of the matrix orthogonality. We also prove that our systems of polynomials are indeed orthonormal with respect to a complex measure of orthogonality. Approximation problems of Hermite-Padé type are also discussed. Finally, a Markov’s type theorem is presented.
ISSN:0167-8019
1572-9036
DOI:10.1007/s10440-010-9577-3