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

Let image be a finite positive Borel measure with compact support consisting of an interval image plus a set of isolated points in image, such that image almost everywhere on image. Let image, be a sequence of polynomials, image, with real coefficients whose zeros lie outside the smallest interval containing the support of image. We prove ratio and relative asymptotics of sequences of orthogonal polynomials with respect to varying measures of the form image. In particular, we obtain an analogue for varying measures of Denisovʹs extension of Rakhmanovʹs 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.