582 research outputs found
Relative asymptotics for orthogonal matrix polynomials
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
A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line
Szego's procedure to connect orthogonal polynomials on the unit circle and
orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It
generates the so-called semi-orthogonal functions on the linear space of
Laurent polynomials L, and leads to a new orthogonality structure in the module
LxL. This structure can be interpreted in terms of a 2x2 matrix measure on
[-1,1], and semi-orthogonal functions provide the corresponding sequence of
orthogonal matrix polynomials. This gives a connection between orthogonal
polynomials on the unit circle and certain classes of matrix orthogonal
polynomials on [-1,1]. As an application, the strong asymptotics of these
matrix orthogonal polynomials is derived, obtaining an explicit expression for
the corresponding Szego's matrix function.Comment: 28 page
Bilinear semi-classical moment functionals and their integral representation
We introduce the notion of bilinear moment functional and study their general
properties. The analogue of Favard's theorem for moment functionals is proven.
The notion of semi-classical bilinear functionals is introduced as a
generalization of the corresponding notion for moment functionals and motivated
by the applications to multi-matrix random models. Integral representations of
such functionals are derived and shown to be linearly independent.Comment: 25 pages, 3 figures, minor correction and change to Figure
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