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 b4f2084704">, with b42744b77fc88f"> 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 b42744b77fc88f">. In this paper we consider the space, denoted by b4053c26213c45e712bcc6d4" 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 b44cf8ee0" title="Click to view the MathML source">gp between the metric subspaces b4053c26213c45e712bcc6d4" title="Click to view the MathML source">Np and Vp, where Vp denotes the space of nontrivial probability measures with associated p -periodic Verblunsky coefficients. Moreover, it is shown that the set a4abc463630ffd841d45" title="Click to view the MathML source">Fp of fixed points of b44cf8ee0" title="Click to view the MathML source">gp is exactly Vp∩Np and this set is characterized by a a4d21ea1cc37d71d6c8fcd01b" title="Click to view the MathML source">(p−1)-dimensional submanifold of Rp. We also prove that the study of probability measures in b4053c26213c45e712bcc6d4" title="Click to view the MathML source">Np is equivalent to the study of probability measures in Vp. Furthermore, it is shown that the pure points of measures in b4053c26213c45e712bcc6d4" 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.