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

Let 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 ′> 0 almost everywhere on [c, d]. Let {w2n}, n ∈ Z+, be a sequence of polynomials, degw2n 2n, with real coefficients whose zeros lie outside the smallest interval containing the support of . We prove ratio and relative asymptotics of sequences of orthogonal polynomials with respect to varying measures of the form d /w2n. 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 ∗Corresponding author. E-mail addresses: [email protected] (D.B. Rolanía), [email protected] (B. de la Calle Ysern), [email protected] (G.L. Lagomasino). 1Research of D.B. Rolanía was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grant BFM 2003-06335-C03-02. 2Research 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. 3Research 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. on the unit circle and for multi-orthogonal polynomials in which the measures involved are of the type described above.