Convex linear combinations of sequences of monic orthogonal polynomials

Abstract

For a sequence { Φ n } 0 ∞ \{\Phi _n\}_0^\infty of monic orthogonal polynomials (SMOP), with respect to a positive measure supported on the unit circle, we obtain necessary and sufficient conditions on a SMOP { Q n } 0 ∞ \{Q_n\}_0^\infty in order that a convex linear combination { R n } 0 ∞ \{R_n\}_0^\infty with R n = β Φ n + ( 1 − β ) Q n R_n=\beta \Phi _n+(1-\beta )Q_n be a SMOP with respect to a positive measure supported on the unit circle.