Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

Authors:
A. I. Aptekarev;J. S. Geronimo;W. Van Assche
Affiliations:
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Square 4, 125047 Moscow, Russia;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA;Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
Venue:
Journal of Approximation Theory
Year:
2008

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Abstract

We consider orthogonal polynomials {P"n","N,n=0,1,2,...}, where n is the degree of the polynomial and N is a discrete parameter. These polynomials are orthogonal with respect to a varying weight W"N which depends on the parameter N and they satisfy a recurrence relation with varying recurrence coefficients which we assume to be varying monotonically as N tends to infinity. We establish the existence of the limit W=lim"N"-"~W"N^1^/^N and link this limit to an external field for an equilibrium problem in logarithmic potential theory.