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 935d65967b5b42744b77fc88f"> 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 9fd"> are given by the relation
where ρ0=1, 94d4f92f3e7f2d6cb8b526d62fa6d">, e721e22aed714cdf2" title="Click to view the MathML source">n≥1 and 94" class="mathmlsrc">94.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=bb132d6a41ce02c224f0c250c51e6f61">94.gif"> is the minimal parameter sequence of 935d65967b5b42744b77fc88f">. 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 and e7c2e4e0bbef0c8"> 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 9385f16d5e" title="Click to view the MathML source">Vp, where 9385f16d5e" title="Click to view the MathML source">Vp denotes the space of nontrivial probability measures with associated p -periodic Verblunsky coefficients. Moreover, it is shown that the set Fp of fixed points of gp is exactly Vp∩Np and this set is characterized by a (p−1)-dimensional submanifold of 8c5a9e1" title="Click to view the MathML source">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 9385f16d5e" title="Click to view the MathML source">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 and e7c2e4e0bbef0c8"> are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.