On the orthogonality of the derivative of the reciprocal sequence

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:157

Article

Suárez, C

关键词: orthogonal polynomials;    semiclassical orthogonal polynomials;    linear regular functionals;    measures on the unit circle;   

DOI  :  10.1016/S0377-0427(03)00384-4

来源: Elsevier

【 摘 要 】

Let {Phi(n)} be a monic orthogonal polynomial sequence on the unit circle (MOPS). The study of the orthogonality properties of the derivative sequence {Phi(n+1)'/(n + 1)} is a classic problem of the orthogonal polynomials theory. In fact, it is well known that the derivative sequence is again a MOPS if and only if Phi(n)(z) = z(n). A similar problem can be posed in terms of the reciprocal sequence of {Phi(n)} as follows:If Phi(n+1)(0) not equal 0, we can define the monic sequence {P-n} by P-n(z) = (Phi(n+1)(*))'(z)/(n + 1)Phin+1(0) n is an element of N = {0, 1,...}, where Phi(n)(*) denotes the reciprocal polynomial of Phi(n), and to study their orthogonality conditions. In this paper we obtain a necessary and sufficient condition for the regularity of {P-n} when the first reflection coefficient Phi(1)(0) is a real number. Also, we give an explicit representation for {Phi(n)} and {P-n}. Moreover, we analyse some questions concerning to the associated functionals of them sequences and the positive definite and semiclassical character. (C) 2003 Published by Elsevier B.V.

【 授权许可】

Free