Perturbation of orthogonal polynomials on an arc of the unit circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg\H{o} recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < |a| < 1. The polynomials then live essentially on the arc {e^(i theta): alpha <= theta <= 2 pi - alpha} where cos alpha/2 = sqrt(1-|a|^2) with 0 <= alpha <= 2 pi. We analyze the orthogonal polynomials by comparing them with the orthogonal polynomials with constant reflection coefficients, which were studied earlier by Ya. L. Geronimus and N. I. Akhiezer. In particular, we show that under certain assumptions on the rate of convergence of the reflection coefficients the orthogonality measure will be absolutely continuous on the arc. In addition, we also prove the unit circle analogue of M. G. Krein's characterization of compactly supported nonnegative Borel measures on the real line whose support contains one single limit point in terms of the corresponding system of orthogonal polynomials

Mean convergence of Lagrange interpolation, II

AbstractThe purpose of the paper is to investigate weighted Lp convergence of Lagrange interpolation taken at the zeros of Hermite polynomials. It is shown that if a continuous function satisfies some growth conditions, then the corresponding Lagrange interpolation process converges in every Lp (1 < p < ∞) provided that the weight function is chosen in a suitable way