Orthogonal Polynomials and Quadratic Transformations

Abstract

Starting from a sequence #left brace#P_n#right brace#n#>=#o of monic polynomials orthogonal with respect to a linear functional #mu#, we find a linear functional#nu# such that Q_n n#>=#o, with either Q_2_n(#chi# P_n(T(#chi#)) or Q_2_n+1(#chi#) (#chi# - #alpha#)P_n(T(#chi#) where T is a monic quadratic polynomial and #alpha# a complex number, is a sequence of monic orthogonal polynomials with respect to #nu#. In particular, we discuss the case when #mu# and #nu# are both positive definite linear functionals. Thus, we obtain a solution for an inverse problem which is a converse, for quadratic mappings, of one analyzed by J.Geronimo and W.Van Assche ("Orthogonal Polynomials on Several Intervals via a Polynomial Mapping, Trans. Amer. Math. Soc., 308 (2),1988,559-581)Available from Departamento de Matematica, Universidade de Coimbra, 3000 Coimbra, Portugal / FCT - Fundação para o Ciência e a TecnologiaSIGLEPTPortuga