In this paper, we exhibit explicitly a sequence of 2 × 2 matrix valued orthogonal polynomials with respect to a weight Wp,n, for any pair of real numbers p and n such that 0 < p < n. The entries of these polynomiales are expressed in terms of the Gegenbauer polynomials C λ k . Also the corresponding three-term recursion relations are given and we make some studies of the algebra of differential operators associated with the weight Wp,n.2010 Mathematics Subject Classification. 22E45 -33C45 -33C47.

The subject of time-band-limiting, originating in signal processing, is dominated by the miracle that a naturally appearing integral operator admits a commuting differential one allowing for a numerically efficient way to compute its eigenfunctions. Bispectrality is an effort to dig into the reasons behind this miracle and goes back to joint work with H. Duistermaat. This search has revealed unexpected connections with several parts of mathematics, including integrable systems.Here we consider a matrix valued version of bispectrality and give a general condition under which we can display a constructive and simple way to obtain the commuting differential operator. Furthermore, we build an operator that commutes with both the time-limiting operator and the band-limiting operators.

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.

Abstract. 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.

In this paper we discuss the notion of reducibility for matrix weights and introduce a real vector space C R which encodes all information about the reducibility of W . In particular a weight W reduces if and only if there is a non-scalar matrix T such that T W = W T * . Also, we prove that reducibility can be studied by looking at the commutant of the monic orthogonal polynomials or by looking at the coefficients of the corresponding three term recursion relation. A matrix weight may not be expressible as direct sum of irreducible weights, but it is always equivalent to a direct sum of irreducible weights. We also establish that the decompositions of two equivalent weights as sums of irreducible weights have the same number of terms and that, up to a permutation, they are equivalent.We consider the algebra of right-hand-side matrix differential operators D(W ) of a reducible weight W , giving its general structure. Finally, we make a change of emphasis by considering reducibility of polynomials, instead of reducibility of matrix weights.2010 Mathematics Subject Classification. 42C05-47L80-33C45. Key words and phrases. Matrix orthogonal polynomials, reducible weights, complete reducibility, the algebra of a reducible weight.