arXiv subjects
Publications and source records attributed to Ignacio Nahuel Zurrián.
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.
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.
In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=({\mathrm{SO}}(n+1),{\mathrm{SO}}(n))$ 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 $D$.
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 $π\in\hat\SO(n)$ if and only if it is an irreducible spherical function of some type $γ\in\hat {\mathrm{O}}(n)$. 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.