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 a935d65967b5b42744b77fc88f"> 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 b8f1772d" title="Click to view the MathML source">ρ0=1, b8b526d62fa6d">, 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 e6f61"> is the minimal parameter sequence of a935d65967b5b42744b77fc88f">. In this paper we consider the space, denoted by Np, of all nontrivial probability measures such that the associated real sequences and a9636849e18783c1e7c2e4e0bbef0c8"> are periodic with period p , for b82d41dd2132bfb22bea267bc698b3" 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 gp between the metric subspaces Np and baa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">Vp, where baa018d8f3fc632afdc589385f16d5e" 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 a9e1" 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 baa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">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 a9636849e18783c1e7c2e4e0bbef0c8"> are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.