@MISC{Measures_representationsof, author = {For Modified Measures}, title = {Representations of Orthogonal Polynomials}, year = {} }

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Abstract

Xin Li\Lambda and F. Marcell'an Abstract. Let d_(`) be a (positive or signed) measure on [0; 2ss] and let g(`) be a ratio of two trigonometric polynomials. Under fairly general condition, we find a relation between polynomials f'n(z; d_)g and f'n(z; gd_)g that are orthogonal with respect to d_(`) and g(`)d_(`), respectively, on the unit circle in the complex plane. This result extends a classical formula of Christoffel and its generalization given by Uvarov for polynomials orthogonal with respect to positive measures on the real line. AMS classification: 15A03, 42C05. Key words and phrases: Basis, orthogonal polynomials, modification of measure. 1. Introduction For a finite positive measure d_(`) on [0; 2ss] with infinitely many points in its support, let fOEn(z; d_)g1n=0 denote the orthonormal polynomials associated with d_(`) on the unit circle, that is, OEn(z; d_) = ^n(d_)zn+ lower degree terms, ^n(d_) ? 0, and