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

34 pages, no figures.-- MSC1991 codes: 42C05, 41A28.-- Dedicated to Barry Simon on the occasion of his sixtieth birthday.MR#: MR2220040 (2006m:42002)Zbl#: Zbl 1100.42014Let 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 $\{w_{2n}\}$, $n\in\Bbb Z_+$, be a sequence of polynomials, $\deg w_{2n}\leq2n$, 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 $\frac{d\mu_n}{w_{2n}}$. 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.Research of D.B. Rolanía was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grant BFM 2003-06335-C03-02. Research of de la Calle Ysern was supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grants BFM 2002-04315-C02-01 and BFM 2003-06335-C03-02. Research of G.L. Lagomasino was supported by Grants INTAS 03-516637, NATO PST.CLG.979738, and by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grant BFM 2003-06335-C03-02.Publicad