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 b296f671647cf0cfcf34f32ecb9fd"> are given by the relation
where b23b5ebcf9d4f3365601b2db8f1772d" title="Click to view the MathML source">ρ0=1, , a197fe721e22aed714cdf2" title="Click to view the MathML source">n≥1 and e6f61"> is the minimal parameter sequence of . In this paper we consider the space, denoted by ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">Np, of all nontrivial probability measures such that the associated real sequences b2926c0e11ab3162da16324f1a061ee1"> and e4e0bbef0c8"> are periodic with period p , for b22bea267bc698b3" 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 ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">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 a1" title="Click to view the MathML source">Vp∩Np and this set is characterized by a a1cc37d71d6c8fcd01b" title="Click to view the MathML source">(p−1)-dimensional submanifold of b2ce8c5a9e1" title="Click to view the MathML source">Rp. We also prove that the study of probability measures in ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">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 ac4f18dcb4053c26213c45e712bcc6d4" 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 b2926c0e11ab3162da16324f1a061ee1"> and e4e0bbef0c8"> are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.