28 research outputs found
The Algebra of Differential Operators for a Gegenbauer Weight Matrix
In this paper we study in detail algebraic properties of the algebra
of differential operators associated to a matrix weight of
Gegenbauer type. We prove that two second order operators generate the algebra,
indeed is isomorphic to the free algebra generated by two
elements subject to certain relations. Also, the center is isomorphic to the
affine algebra of a singular rational curve. The algebra is a
finitely-generated torsion-free module over its center, but it is not flat and
therefore it is not projective.
This is the second detailed study of an algebra and the first
one coming from spherical functions and group representations. We prove that
the algebras for different Gegenbauer weights and the algebras studied
previously, related to Hermite weights, are isomorphic to each other. We give
some general results that allow us to regard the algebra as the
centralizer of its center in the Weyl algebra. We do believe that this should
hold for any irreducible weight and the case considered in this paper
represents a good step in this direction
Matrix Gegenbauer Polynomials: the Fundamental Cases
In this paper, we exhibit explicitly a sequence of matrix valued
orthogonal polynomials with respect to a weight , for any pair of real
numbers and such that .
The entries of these polynomiales are expressed in terms of the Gegenbauer
polynomials . Also the corresponding three-term recursion
relations are given and we make some studies of the algebra of differential
operators associated with the weight
Bispectrality for Matrix Laguerre-Sobolev polynomials
In this contribution we deal with sequences of polynomials orthogonal with
respect to a Sobolev type inner product. A banded symmetric operator is
associated with such a sequence of polynomials according to the higher order
difference equation they satisfy. Taking into account the Darboux
transformation of the corresponding matrix we deduce the connection with a
sequence of orthogonal polynomials associated with a Christoffel perturbation
of the measure involved in the standard part of the Sobolev inner product. A
connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev
type case is studied as an illustrative example. Finally, the bispectrality of
such matrix orthogonal polynomials is pointed out
Time and band limiting for matrix valued functions: an integral and a commuting differential operator
The problem of recovering a signal of finite duration from a piece of its
Fourier transform was solved at Bell Labs in the 's, by exploiting a
"miracle": a certain naturally appearing integral operator commutes with an
explicit differential one. Here we show that this same miracle holds in a
matrix valued version of the same problem
Spherical Functions: The Spheres Vs. The Projective Spaces
In this paper we establish a close relationship between the spherical functions of the n-dimensional sphere S^n\simeq\SO(n+1)/\SO(n) and the spherical functions of the n-dimensional real projective space P^n(\mathbb{R})\simeq\SO(n+1)/\mathrm{O}(n). In fact, for n odd a function on \SO(n+1) is an irreducible spherical function of some type \pi\in\hat\SO(n) if and only if it is an irreducible spherical function of some type γ∈O^(n). When n is even this is also true for certain types, and in the other cases we exhibit a clear correspondence between the irreducible spherical functions of both pairs (\SO(n+1),\SO(n)) and (\SO(n+1),\mathrm{O}(n)). Summarizing, to find all spherical functions of one pair is equivalent to do so for the other pair.Fil: Tirao, Juan Alfredo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin
Spherical functions of fundamental K-types associated with the n-dimensional sphere
In this paper, we describe the irreducible spherical functions of fundamental K-types associated with the pair (G, K) = (SO(n + 1), SO(n)) in terms of matrix hypergeometric functions. The output of this description is that the irreducible spherical functions of the same K-fundamental type are encoded in new examples of classical sequences of matrix-valued orthogonal polynomials, of size 2 and 3, with respect to a matrix-weight W supported on [0, 1]. Moreover, we show that W has a second order symmetric hypergeometric operator D.publishedVersionFil: Tirao, Juan Alfredo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Tirao, Juan Alfredo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Zurrián, Ignacio Nahuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.Matemática Pur
Bispectrality and Time-Band-Limiting: Matrix valued polynomials
The subject of time-band-limiting, originating in signal processing, is dominated by the miracle that a naturallyappearing integral operator admits a commuting dierential one, allowing for a numerically ecient way to compute itseigenfunctions. Bispectrality is an eort to dig into the reasons behind this miracle and goes back to joint work with H.Duistermaat. This search has revealed unexpected connections with several parts of mathematics, including integrablesystems. Here we consider a matrix-valued version of bispectrality and give a general condition under which we candisplay a constructive and simple way to obtain the commuting dierential operator. Furthermore, we build an operatorthat commutes with both the time-limiting operator and the band-limiting operators.Fil: Grünbaum, Alberto. University of California at Berkeley; Estados UnidosFil: Pacharoni, Maria Ines. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin