Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle

详细信息    查看全文

• 作者：Cleonice F. Bracciali^a ; ^{cleonice@ibilce.unesp.br" class="auth_mail" title="E-mail the corresponding author} ; Jairo S. Silva^b ; ^c ; ^{jairo.santos@ufma.br" class="auth_mail" title="E-mail the corresponding author} ; A. Sri Ranga^a ; ^{ranga@ibilce.unesp.br" class="auth_mail" title="E-mail the corresponding author} ; Daniel O. Veronese^d ; ^c ; ^{veronese@icte.uftm.edu.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Probability measures ; Periodic Verblunsky coefficients ; Chain sequences ; Periodic real sequences
• 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 ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=c00fdb7b599c8631f42bd9b4f2084704">, with ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f"> 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 ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b36b296f671647cf0cfcf34f32ecb9fd"> are given by the relation where ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=5b23b5ebcf9d4f3365601b2db8f1772d" title="Click to view the MathML source">ρ₀=1${\rho }_0=1$, ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=83d94d4f92f3e7f2d6cb8b526d62fa6d">, ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=76424479c0a197fe721e22aed714cdf2" title="Click to view the MathML source">n≥1$n\ge 1$ and ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=bb132d6a41ce02c224f0c250c51e6f61"> is the minimal parameter sequence of ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f">. In this paper we consider the space, denoted by ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_p$N_p$, of all nontrivial probability measures such that the associated real sequences ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1"> and ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8"> are periodic with period p  , for ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=38b82d41dd2132bfb22bea267bc698b3" title="Click to view the MathML source">p∈N$p\in \mathbb{N}$. By assuming an appropriate metric on the space of all nontrivial probability measures on the unit circle, we show that there exists a homeomorphism ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">g_p$g_p$ between the metric subspaces ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_p$N_p$ and ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_p$V_p$, where ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_p$V_p$ denotes the space of nontrivial probability measures with associated p  -periodic Verblunsky coefficients. Moreover, it is shown that the set ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=8838092044eda4abc463630ffd841d45" title="Click to view the MathML source">F_p$F_p$ of fixed points of ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">g_p$g_p$ is exactly ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=d00c29731cc554f4abc6356216db67a1" title="Click to view the MathML source">V_p∩N_p$V_p\cap N_p$ and this set is characterized by a ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=aab7e30a4d21ea1cc37d71d6c8fcd01b" title="Click to view the MathML source">(p−1)-dimensional submanifold of ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=af1316156a55e2691d427b2ce8c5a9e1" title="Click to view the MathML source">R^p${\mathbb{R}}^p$. We also prove that the study of probability measures in ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_p$N_p$ is equivalent to the study of probability measures in ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_p$V_p$. Furthermore, it is shown that the pure points of measures in ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_p$N_p$ 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 ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1"> and ser=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8"> are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.