Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures

Abstract

Let μ be a finite positive Borel measure with compact support consisting of an interval [ c , d ] ⊂ R plus a set of isolated points in R ⧹ [ c , d ] , such that μ ′ > 0 almost everywhere on [ c , d ] . Let { w 2 n } , n ∈ Z + , be a sequence of polynomials, deg w 2 n ⩽ 2 n , with real coefficients whose zeros lie outside the smallest interval containing the support of μ . We prove ratio and relative asymptotics of sequences of orthogonal polynomials with respect to varying measures of the form d μ / w 2 n . In particular, we obtain an analogue for varying measures of Denisov's extension of Rakhmanov's theorem on ratio asymptotics. These results on varying measures are applied to obtain ratio asymptotics for orthogonal polynomials with respect to fixed measures on the unit circle and for multi-orthogonal polynomials in which the measures involved are of the type described above.