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, 92f3e7f2d6cb8b526d62fa6d">, 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 e712bcc6d4" title="Click to view the MathML source">Np, of all nontrivial probability measures such that the associated real sequences 926c0e11ab3162da16324f1a061ee1"> 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 e712bcc6d4" title="Click to view the MathML source">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 92044eda4abc463630ffd841d45" 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 Rp. We also prove that the study of probability measures in e712bcc6d4" title="Click to view the MathML source">Np is equivalent to the study of probability measures in Vp. Furthermore, it is shown that the pure points of measures in e712bcc6d4" title="Click to view the MathML source">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 926c0e11ab3162da16324f1a061ee1"> and 849e18783c1e7c2e4e0bbef0c8"> are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.