Orthogonal polynomials and jump modifications

Abstract

Let $\left\{ {\hat P_n \left( {z,d\alpha } \right)} \right\}$ be a system of orthonormal polynomials on the unit circle U with respect to a measure α and let $\left\{ {\hat P_n \left( {z,d\beta } \right)} \right\}$ be a system of orthonormal polynomials on U with respect to β (β=α+εδ(t), where t≠U, ε>0, and δ(t) the unit measure supported at z=t). If (e_n (α))^−1/2 and (e_n (β))^−1/2 are their leading coefficients, in this paper we present some properties of the difference e_n (β)-e_n(α) and we show that both se quences e_n (α) and e_n (β) have the same limit by using the recurrence re lation.