刊名:Journal of Mathematical Analysis and Applications
出版年:2017
出版时间:1 January 2017
年:2017
卷:445
期:1
页码:719-745
全文大小:672 K
文摘
It is known that given a pair of real sequences 9c8631f42bd9b4f2084704">, with a positive chain sequence, we can associate a unique nontrivial probability measure μ on the unit circle. Precisely, the measure is such that the corresponding Verblunsky coefficients are given by the relation
where ρ0=1, , 9c0a197fe721e22aed714cdf2" title="Click to view the MathML source">n≥1 and bb132d6a41ce02c224f0c250c51e6f61"> is the minimal parameter sequence of . In this paper we consider the space, denoted by Np, of all nontrivial probability measures such that the associated real sequences and 849e18783c1e7c2e4e0bbef0c8"> are periodic with period p , for p∈N. By assuming an appropriate metric on the space of all nontrivial probability measures on the unit circle, we show that there exists a homeomorphism gp between the metric subspaces Np and Vp, where Vp denotes the space of nontrivial probability measures with associated p -periodic Verblunsky coefficients. Moreover, it is shown that the set 841d45" title="Click to view the MathML source">Fp of fixed points of gp is exactly Vp∩Np and this set is characterized by a (p−1)-dimensional submanifold of 9e1" title="Click to view the MathML source">Rp. We also prove that the study of probability measures in Np is equivalent to the study of probability measures in Vp. Furthermore, it is shown that the pure points of measures in Np are, in fact, zeros of associated para-orthogonal polynomials of degree p . We also look at the essential support of probability measures in the limit periodic case, i.e., when the sequences and 849e18783c1e7c2e4e0bbef0c8"> are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.