Browsing by Author "Branquinho, A."
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- Dynamics and interpretation of some integrable systems via matrix orthogonal polynomialsPublication . Branquinho, A.; Moreno, A. Foulquié; Mendes, A.In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a com plex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, general ized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattice.
- Matrix interpretation of multiple orthogonalityPublication . Branquinho, A.; Cotrim, L.; Moreno, A. FoulquiéIn this work we give an interpretation of a (s (d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials. We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions. We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite-Padé approximation in matrix form is given.
- Relative asymptotics for orthogonal matrix polynomialsPublication . Branquinho, A.; Marcellán, F.; Mendes, A.In this paper we study sequences of matrix polynomials that satisfy a non-symmetric recurrence relation. To study this kind of sequences we use a vector interpretation of the matrix orthogonality. In the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix Nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class. We study the generalized matrix Chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples. The concept of a Dirac delta functional is introduced. We show how the vector model that includes a Dirac delta functional is a representation of a discrete Sobolev inner product. It also allows to reinterpret such perturbations in the usual matrix Nevai class. Finally, the relative asymptotics between a polynomial in the generalized matrix Nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a Dirac delta functional is deduced. © 2012 Elsevier Ltd. All rights reserved.
- Vector Interpretation of the Matrix Orthogonality on the Real LinePublication . Branquinho, A.; Marcellán, F.; Mendes, A.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.