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 , with 967b5b42744b77fc88f"> 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 96f671647cf0cfcf34f32ecb9fd"> are given by the relation
where ρ0=1, e7f2d6cb8b526d62fa6d">, e721e22aed714cdf2" title="Click to view the MathML source">n≥1 and e6f61"> is the minimal parameter sequence of 967b5b42744b77fc88f">. 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 ab3162da16324f1a061ee1"> and 9636849e18783c1e7c2e4e0bbef0c8"> are periodic with period p , for 8b82d41dd2132bfb22bea267bc698b3" title="Click to view the MathML source">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 8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">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 abc463630ffd841d45" title="Click to view the MathML source">Fp of fixed points of 8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">gp is exactly abc6356216db67a1" title="Click to view the MathML source">Vp∩Np and this set is characterized by a ab7e30a4d21ea1cc37d71d6c8fcd01b" title="Click to view the MathML source">(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 ab3162da16324f1a061ee1"> and 9636849e18783c1e7c2e4e0bbef0c8"> are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.