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Revisiting Biorthogonal Polynomials: An LU Factorization Discussion

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Orthogonal Polynomials: Current Trends and Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 22))

Abstract

The Gauss–Borel or LU factorization of Gram matrices of bilinear forms is the pivotal element in the discussion of the theory of biorthogonal polynomials. The construction of biorthogonal families of polynomials and its second kind functions, of the spectral matrices modeling the multiplication by the independent variable x, the Christoffel–Darboux kernel and its projection properties, are discussed from this point of view. Then, the Hankel case is presented and different properties, specific of this case, as the three terms relations, Heine formulas, Gauss quadrature and the Christoffel–Darboux formula are given. The classical orthogonal polynomial of Hermite, Laguerre and Jacobi type are discussed and characterized within this scheme. Finally, it is shown who this approach is instrumental in the derivation of Christoffel formulas for general Christoffel and Geronimus perturbations of the bilinear forms.

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Acknowledgements

The author thanks the financial support from the Spanish Agencia Estatal de Investigación, research project PGC2018-096504-B-C3 entitled Ortogonalidad y Aproximación: Teoría y Aplicaciones en Física Matemática. The author acknowledges the exhaustive revision by the anonymous referee that much improved the readability of this contribution.

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Authors and Affiliations

  1. Departamento de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain

    Manuel Mañas

  2. Instituto de Ciencias Matematicas (ICMAT), Madrid, Spain

    Manuel Mañas

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  1. Manuel Mañas
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Correspondence to Manuel Mañas .

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Editors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain

    Francisco Marcellán

  2. Departamento de Física y Matemáticas, Universidad de Alcalá, Alcalá de Henares, Madrid, Spain

    Edmundo J. Huertas

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Mañas, M. (2021). Revisiting Biorthogonal Polynomials: An LU Factorization Discussion. In: Marcellán, F., Huertas, E.J. (eds) Orthogonal Polynomials: Current Trends and Applications. SEMA SIMAI Springer Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-56190-1_10

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