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Inés Pacharoni

Publications and source records attributed to Inés Pacharoni.

13 recordsLinked to original sources
arXiv2026

On a new equivalence relation for matrix-valued orthogonal polynomials

In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and how to build such a transformation explicitly. In particular, they allow us to see when and how any given sequence of polynomials is Darboux-related to classic orthogonal polynomials. We also explore the notion of Darboux-irreducibility and study some sequences that are not a Darboux transformation of classical orthogonal polynomials.

math.CA↗
arXiv2025

Structure of operator algebras for matrix orthogonal polynomials

In this paper, we study the structure of the differential operator algebra \( \mathcal{D}(W) \) and its associated eigenvalue algebra \( Λ(W) \) for matrix-valued orthogonal polynomials. While \( Λ(W) \) is isomorphic to \( \mathcal{D}(W) \), its simpler framework allows us to efficiently derive strong results about \( \mathcal{D}(W) \) and its center \( \mathcal{Z}(W) \). We analyze the behavior of the center under Darboux transformations, establishing explicit relationships between the centers of Darboux-equivalent weights. These results are illustrated through the study of both reducible and irreducible matrix weights, including a detailed analysis of an irreducible Jacobi-type weight.

math.CA↗
arXiv2025

Matrix-Valued Hermite and Laguerre polynomials via Quadratic Transformation

We present the first systematic extension of the classical Hermite-Laguerre quadratic correspondence to the matrix-valued setting. Starting from a Hermite-type weight matrix W(x) = exp(-x^2) Z(x) with W(x) = W(-x), the change of variables y = x^2 produces two Laguerre-type weights with parameters alpha = -1/2 and alpha = 1/2, and relates the corresponding sequences of matrix-valued orthogonal polynomials through an explicit decomposition into even and odd subsequences. We prove that this transformation preserves differential operators and Darboux transformations, thereby establishing a direct structural link between the Hermite and Laguerre sides and providing new constructive tools for the matrix Bochner problem. Concrete families - including a new 3x3 example and an arbitrary-size family built from block-nilpotent matrices - illustrate the theory and supply fresh sources of matrix-valued orthogonal polynomials endowed with non-trivial differential algebras.

math.CA↗
arXiv2025

The algebra $\mathcal{D}(W)$ via strong Darboux transformations

The Matrix Bochner Problem aims to classify weight matrices $W$ such that the algebra $\mathcal D(W)$, of all differential operators that have a sequence of matrix-valued orthogonal polynomials for $W$ as eigenfunctions, contains a second-order differential operator. In \cite{CY18} it is proven that, under certain assumptions, the solutions to the Matrix Bochner Problem can be obtained through a noncommutative bispectral Darboux transformation of some classical scalar weights. The main aim of this paper is to introduce the concept of strong Darboux transformation among weight matrices and explore the relationship between the algebras $\mathcal{D}(W)$ and $\mathcal{D}(\widetilde{W})$ when $\widetilde{W}$ is a strong Darboux transformation of $W$. Starting from a direct sum of classical scalar weights $\widetilde W$, and leveraging our complete knowledge of the algebra of $\mathcal D(\widetilde W)$, we can easily determine the algebra $\mathcal D(W)$ of a weight $W$ that is a strong Darboux transformation of $\widetilde W$.

math.CA↗
arXiv2025

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

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, and it is closely related to Darboux transformations of some differential operators. This paper aims to study Darboux transformations between weight matrices and to establish a direct connection with the structure of the algebra $\mathcal D(W)$ of all differential operators that have a sequence of matrix-valued orthogonal polynomials with respect to $W$ as eigenfunctions.

math.CA↗
arXiv2024

Singular solutions of the matrix Bochner problem: the $N$-dimensional cases

In the theory of matrix-valued orthogonal polynomials, there exists a longstanding problem known as the Matrix Bochner Problem: the classification of all $N \times N$ weight matrices $W(x)$ such that the associated orthogonal polynomials are eigenfunctions of a second-order differential operator. In [4], Casper and Yakimov made an important breakthrough in this area, proving that, under certain hypotheses, every solution to this problem can be obtained as a bispectral Darboux transformation of a direct sum of classical scalar weights. In the present paper, we construct three families of weight matrices $W(x)$ of size $N \times N$, associated with Hermite, Laguerre, and Jacobi weights, which can be considered 'singular' solutions to the Matrix Bochner Problem because they cannot be obtained as a Darboux transformation of classical scalar weights.

math.CA↗
arXiv2024

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 [4], 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.

math.CA↗
arXiv2018

Mehler-Heine formula: a generalization in the context of spherical functions

In this article, using the notion of group contraction, we obtain the spherical functions of the strong Gelfand pair $(\mathrm{M}(n),\mathrm{SO}(n))$ as an appropriate limit of spherical functions of the strong Gelfand pair $(\mathrm{SO}(n+1),\mathrm{SO}(n))$ and also of the strong Gelfand pair $(\mathrm{SO}_0(n,1),\mathrm{SO}(n))$.

math.RT↗
arXiv2018

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 naturally appearing integral operator admits a commuting differential one allowing for a numerically efficient way to compute its eigenfunctions. Bispectrality is an effort 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 integrable systems. Here we consider a matrix valued version of bispectrality and give a general condition under which we can display a constructive and simple way to obtain the commuting differential operator. Furthermore, we build an operator that commutes with both the time-limiting operator and the band-limiting operators.

math.CA↗
arXiv2016

Matrix Gegenbauer Polynomials: the $2\times 2$ 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 $C_k^λ$. Also the corresponding three-term recursion relations are given and we make some studies of the algebra of differential operators associated with the weight $W_{p,n}$.

math.RT↗
arXiv2015

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.

math.SP↗
arXiv2013

Spherical Functions Associated With the Three Dimensional Sphere

In this paper, we determine all irreducible spherical functions Φof any K -type associated to the pair (G,K)=(\SO(4),\SO(3)). This is accomplished by associating to Φa 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 π_\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.

math.CA↗