Time and Band Limiting for Matrix Valued Functions, an Example

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.

References in corpus (5)

• Spatiospectral concentration on a sphere
• Spherical Slepian functions and the polar gap in geodesy
• Bispectral algebras of commuting ordinary differential operators
• The approximation of almost time and band limited functions by their expansion in some orthogonal polynomials bases
• Spherical Functions of Fundamental $K$-Types Associated with the $n$-Dimensional Sphere

Cited by in corpus (9)

• Matrix Gegenbauer Polynomials: the $2\times 2$ Fundamental Cases
• Bispectrality and Time-Band-Limiting: Matrix valued polynomials
• Time and band limiting for matrix valued functions: an integral and a commuting differential operator
• Augmented Slepians: Bandlimited Functions that Counterbalance Energy in Selected Intervals
• The Algebra of Differential Operators for a Gegenbauer Weight Matrix
• $2\times 2$ Hypergeometric operators with diagonal eigenvalues
• A further look at time-and-band limiting for matrix orthogonal polynomials
• Time and band limiting for exceptional polynomials
• The CMV bispectral problem