Sobolev Orthogonal Polynomials on the Unit Circle and Coherent Pairs of Measures of the Second Kind

Abstract

We refer to a pair of non trivial probability measures (μ0, μ1) supported on the unit circle as a coherent pair of measures of the second kind on the unit circle if the corresponding sequences of monic orthogonal polynomials {Φn(μ0;z)}n≥0 and {Φn(μ1;z)}n≥0 satisfy 1nΦn′(μ0;z)=Φn-1(μ1;z)-χnΦn-2(μ1;z), n≥ 2. It turns out that there are more interesting examples of pairs of measures on the unit circle with this latter coherency property than in the case of the standard coherence. The main objective in this contribution is to determine such pairs of measures. The polynomials orthogonal with respect to the Sobolev inner products associated with coherent pairs of measures of the second kind are also studied.