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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