Asymptotics of orthogonal polynomials for a weight with a jump on [-1; 1]
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2011Resumen
We consider the orthogonal polynomials on [−1,1] with respect to the weight where h is real analytic and strictly positive on [−1,1] and Ξ c is a step-like function: Ξ c (x)=1 for x∈[−1,0) and Ξ c (x)=c 2, c>0, for x∈[0,1]. We obtain strong uniform asymptotics of the monic orthogonal polynomials in , as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n→∞. In particular, we prove for w c a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel-Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior. For the asymptotic analysis we use the steepest descent method of Deift and Zhou applied to the noncommutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The...
Palabra/s clave
Polinomios ortogonales
Asintótica
Análisis de Riemann-Hilbert
Ceros
Comportamiento local
Funciones hipergeométricas confluentes
Universalidad
Espacio de Branges
Orthogonal polynomials
Asymptotics
Riemann-Hilbert analysis
Zeros
Local behavior
Confluent hypergeometric functions
Reproducing kernel
Universality
de Branges spaces