Matrix Gegenbauer Polynomials: the 2×22\times 2 Fundamental Cases

In this paper, we exhibit explicitly a sequence of $2\times2$ matrix valued orthogonal polynomials with respect to a weight $W_{p,n}$, for any pair of real numbers $p$ and $n$ such that $0<p<n$. The entries of these polynomiales are expressed in terms of the Gegenbauer polynomials $C_k^\lambda$. Also the corresponding three-term recursion relations are given and we make some studies of the algebra of differential operators associated with the weight $W_{p,n}$

Darboux transformations and the algebra D(W)\mathcal{D}(W)

The problem of finding weight matrices $W(x)$ of size $N \times N$ such that the associated sequence of matrix-valued orthogonal polynomials are eigenfunctions of a second-order matrix differential operator is known as the Matrix Bochner Problem. This paper aims to study Darboux transformations between these weight matrices and to establish a direct connection with the structure of the algebra $\mathcal D(W)$. We find several general properties and provide an explicit description of the Darboux equivalence classes for scalar classical weights. Additionally, we determine the algebra $\mathcal{D}(W)$ when $W$ is a direct sum of classical scalar weights.Comment: 21 page

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 $1960$'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

Singular examples of the Matrix Bochner Problem

The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in [2], demonstrated that, under certain hypotheses, all solutions to the Matrix Bochner Problem are noncommutative bispectral Darboux transformations of a direct sum of classical scalar weights. This paper aims to provide the first proof that there are solutions to the Matrix Bochner Problem that do not arise through a noncommutative bispectral Darboux transformation of any direct sum of classical scalar weights. This initial example could contribute to a more comprehensive understanding of the general solution to the Matrix Bochner Problem.Comment: 19 page

Time and band limiting for matrix valued functions, an example

The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of “time and band limiting” admits a commuting local operator. This is a noncommutative analog of the famous prolate spheroidal wave operator.http://www.emis.de/journals/SIGMA/2015/044/sigma15-044.pdfpublishedVersionFil: Grünbaum, Francisco Alberto. University of California, Berkeley. Department of Mathematics; United States of America.Fil: Pacharoni, María Inés. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Pacharoni, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Pacharoni, María Inés. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil:Zurrián, Ignacio Nahuel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Zurrián, Ignacio Nahuel. 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.Matemática Pur

Time and band limiting for matrix valued functions : an integral and a commuting differential operator

Artículo finalmente publicado en: Grünbaum, F. A., Pacharoni, M. I. y Zurrián, I. N. (2017). Time and band limiting for matrix valued functions : an integral and a commuting differential operator. Inverse Problems, 33 (2), 025005. https://doi.org/10.1088/1361-6420/aa53b8Fil: Grünbaum, Francisco Alberto. University of California, Berkeley. Department of Mathematics; United States of America.Fil: Pacharoni, María Inés. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.Fil: Pacharoni, María Inés. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Pacharoni, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Zurrián, Ignacio Nahuel. Pontificia Universidad Católica de Chile. Facultad de Matematicas; Chile.The problem of recovering a signal of finite duration from a piece of its Fourier transform was solved at Bell Labs in the 1960’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.This research was supported in part by CONICET grant PIP 112-200801-01533, SeCyT-UNC, FONDECYT 3160646 and FA9550-16-1-0175.http://iopscience.iop.org/article/10.1088/1361-6420/aa53b8info:eu-repo/semantics/acceptedVersionFil: Grünbaum, Francisco Alberto. University of California, Berkeley. Department of Mathematics; United States of America.Fil: Pacharoni, María Inés. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.Fil: Pacharoni, María Inés. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Pacharoni, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Zurrián, Ignacio Nahuel. Pontificia Universidad Católica de Chile. Facultad de Matematicas; Chile.Matemática Pur

Spherical Functions Associated With the Three Dimensional Sphere

In this paper, we determine all irreducible spherical functions \Phi of any K -type associated to the pair (G,K)=(\SO(4),\SO(3)). This is accomplished by associating to \Phi a vector valued function H=H(u) of a real variable u, which is analytic at u=0 and whose components are solutions of two coupled systems of ordinary differential equations. By an appropriate conjugation involving Hahn polynomials we uncouple one of the systems. Then this is taken to an uncoupled system of hypergeometric equations, leading to a vector valued solution P=P(u) whose entries are Gegenbauer's polynomials. Afterward, we identify those simultaneous solutions and use the representation theory of \SO(4) to characterize all irreducible spherical functions. The functions P=P(u) corresponding to the irreducible spherical functions of a fixed K-type \pi_\ell are appropriately packaged into a sequence of matrix valued polynomials (P_w)_{w\ge0} of size (\ell+1)\times(\ell+1). Finally we proved that \widetilde P_w={P_0}^{-1}P_w is a sequence of matrix orthogonal polynomials with respect to a weight matrix W. Moreover we showed that W admits a second order symmetric hypergeometric operator \widetilde D and a first order symmetric differential operator \widetilde E.Comment: 49 pages, 2 figure

Proyecto, investigación e innovación en urbanismo, arquitectura y diseño industrial

Actas de congresoLas VII Jornadas de Investigación “Encuentro y Reflexión” y I Jornadas de Investigación de becarios y doctorandos. Proyecto, investigación e innovación en Urbanismo, Arquitectura y Diseño Industrial se centraron en cuatro ejes: el proyecto; la dimensión tecnológica y la gestión; la dimensión social y cultural y la enseñanza en Arquitectura, Urbanismo y Diseño Industrial, sustentados en las líneas prioritarias de investigación definidas epistemológicamente en el Consejo Asesor de Ciencia y Tecnología de esta Universidad Nacional de Córdoba. Con el objetivo de afianzar continuidad, formación y transferencia de métodos, metodología y recursos se incorporó becarios y doctorandos de los Institutos de investigación. La Comisión Honoraria la integraron las tres Secretarias de Investigación de la Facultad, arquitectas Marta Polo, quien fundó y María del Carmen Franchello y Nora Gutiérrez Crespo quienes continuaron la tradición de la buena práctica del debate en la cotidianeidad de la propia Facultad. Los textos que conforman las VII Jornadas son los avances y resultados de las investigaciones realizadas en el bienio 2016-2018.Fil: Novello, María Alejandra. Universidad Nacional de Córdoba. Facultad de Arquitectura, Urbanismo y Diseño; ArgentinaFil: Repiso, Luciana. Universidad Nacional de Córdoba. Facultad de Arquitectura, Urbanismo y Diseño; ArgentinaFil: Mir, Guillermo. Universidad Nacional de Córdoba. Facultad de Arquitectura, Urbanismo y Diseño; ArgentinaFil: Brizuela, Natalia. Universidad Nacional de Córdoba. Facultad de Arquitectura, Urbanismo y Diseño; ArgentinaFil: Herrera, Fernanda. Universidad Nacional de Córdoba. Facultad de Arquitectura, Urbanismo y Diseño; ArgentinaFil: Períes, Lucas. Universidad Nacional de Córdoba. Facultad de Arquitectura, Urbanismo y Diseño; ArgentinaFil: Romo, Claudia. Universidad Nacional de Córdoba. Facultad de Arquitectura, Urbanismo y Diseño; ArgentinaFil: Gordillo, Natalia. Universidad Nacional de Córdoba. Facultad de Arquitectura, Urbanismo y Diseño; ArgentinaFil: Andrade, Elena Beatriz. Universidad Nacional de Córdoba. Facultad de Arquitectura, Urbanismo y Diseño; Argentin