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 db7b599c8631f42bd9b4f2084704">, 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 71647cf0cfcf34f32ecb9fd"> are given by the relation
where db8f1772d" title="Click to view the MathML source">ρ0=1, , 714cdf2" 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 712bcc6d4" title="Click to view the MathML source">Np, of all nontrivial probability measures such that the associated real sequences and 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 712bcc6d4" title="Click to view the MathML source">Np and dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">Vp, where dbaa018d8f3fc632afdc589385f16d5e" 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 db67a1" title="Click to view the MathML source">Vp∩Np and this set is characterized by a 71d6c8fcd01b" title="Click to view the MathML source">(p−1)-dimensional submanifold of Rp. We also prove that the study of probability measures in 712bcc6d4" title="Click to view the MathML source">Np is equivalent to the study of probability measures in dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">Vp. Furthermore, it is shown that the pure points of measures in 712bcc6d4" 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 are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.