### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 044, 14 pages      arXiv:1410.1232      https://doi.org/10.3842/SIGMA.2015.044
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

### Time and Band Limiting for Matrix Valued Functions, an Example

F. Alberto Grünbaum a, Inés Pacharoni b and Ignacio Nahuel Zurrián b
a) Department of Mathematics, University of California, Berkeley 94705, USA
b) CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina

Received February 11, 2015, in final form May 30, 2015; Published online June 12, 2015

Abstract
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.

Key words: time-band limiting; double concentration; matrix valued orthogonal polynomials.

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