54 research outputs found
Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials
Spectral and factorization properties of oscillatory matrices lead to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue–Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is givenpublishe
Matrix differential equations and scalar polynomials satisfying higher order recursions
We show that any scalar differential operator with a family of polyno- mials
as its common eigenfunctions leads canonically to a matrix differen- tial
operator with the same property. The construction of the correspond- ing family
of matrix valued polynomials has been studied in [D1, D2, DV] but the existence
of a differential operator having them as common eigen- functions had not been
considered This correspondence goes only one way and most matrix valued
situations do not arise in this fashion.
We illustrate this general construction with a few examples. In the case of
some families of scalar valued polynomials introduced in [GH] we take a first
look at the algebra of all matrix differential operators that share these
common eigenfunctions and uncover a number of phenomena that are new to the
matrix valued case
On the full Kostant-Toda system and the discrete Korteweg-de Vries equations
The relation between the solutions of the full Kostant–Toda lattice and the discrete Korteweg–de Vries equation is analyzed. A method for constructing solutions of these systems is given. As a consequence of the matricial interpretation of this method, the transform of Darboux is extended for general Hessenberg banded matrices
The CMV bispectral problem
A classical result due to Bochner classifies the orthogonal polynomials on
the real line which are common eigenfunctions of a second order linear
differential operator. We settle a natural version of the Bochner problem on
the unit circle which answers a similar question concerning orthogonal Laurent
polynomials and can be formulated as a bispectral problem involving CMV
matrices. We solve this CMV bispectral problem in great generality proving
that, except the Lebesgue measure, no other one on the unit circle yields a
sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear
differential operator of arbitrary order. Actually, we prove that this is the
case even if such an eigenfunction condition is imposed up to finitely many
orthogonal Laurent polynomials.Comment: 25 pages, final version, to appear in International Mathematics
Research Notice
A new commutativity property of exceptional orthogonal polynomials
We exhibit three examples showing that the "time-and-band limiting"
commutative property found and exploited by D. Slepian, H. Landau and H. Pollak
at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy
and H. Widom in Random matrix theory, holds for exceptional orthogonal
polynomials. The property in question is the existence of local operators with
simple spectrum that commute with naturally appearing global ones. We
illustrate numerically the advantage of having such a local operator
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