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 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, e7f2d6cb8b526d62fa6d">, 90" class="mathmlsrc">90.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=76424479c0a197fe721e22aed714cdf2" title="Click to view the MathML source">n≥1 and is the minimal parameter sequence of . In this paper we consider the space, denoted by 8dcb4053c26213c45e712bcc6d4" 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 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 8dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">Np and 8d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">Vp, where 8d8f3fc632afdc589385f16d5e" 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 8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">gp is exactly Vp∩Np and this set is characterized by a (p−1)-dimensional submanifold of e8c5a9e1" title="Click to view the MathML source">Rp. We also prove that the study of probability measures in 8dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">Np is equivalent to the study of probability measures in 8d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">Vp. Furthermore, it is shown that the pure points of measures in 8dcb4053c26213c45e712bcc6d4" 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.