The Darboux process and time-and-band limiting for matrix orthogonal polynomials

We extend to a situation involving matrix valued orthogonal polynomials a scalar result that originates in work of Claude Shannon in laying the mathematical foundations of information theory and a remarkable series of papers by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's. We show that in this case an algebraic miracle that plays a very important role in the classical case survives an application of the so-called Darboux process in the matrix valued context.