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

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• 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 88&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=c00fdb7b599c8631f42bd9b4f2084704">88-si1.gif">$\{{\{c_n\}}_{n=1}^{\infty },{\{d_n\}}_{n=1}^{\infty }\}$, with 88&_mathId=si101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f">88-si101.gif">${\{d_n\}}_{n=1}^{\infty }$ 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 88&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b36b296f671647cf0cfcf34f32ecb9fd">88-si3.gif">${\{{\alpha }_n\}}_{n=0}^{\infty }$ are given by the relation where 88&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=5b23b5ebcf9d4f3365601b2db8f1772d" title="Click to view the MathML source">ρ₀=1${\rho }_0=1$, 88&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=83d94d4f92f3e7f2d6cb8b526d62fa6d">88-si6.gif">${\rho }_n={\prod }_{k=1}^n(1-i{c}_k)/(1+i{c}_k)$, 90" class="mathmlsrc">88&_mathId=si490.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=76424479c0a197fe721e22aed714cdf2" title="Click to view the MathML source">n≥1 and 88&_mathId=si394.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=bb132d6a41ce02c224f0c250c51e6f61">88-si394.gif">${\{m_n\}}_{n=0}^{\infty }$ is the minimal parameter sequence of 88&_mathId=si101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f">88-si101.gif">${\{d_n\}}_{n=1}^{\infty }$. In this paper we consider the space, denoted by 88&_mathId=si10.gif&_user=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 88&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1">88-si11.gif">${\{c_n\}}_{n=1}^{\infty }$ and 88&_mathId=si12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8">88-si12.gif">${\{m_n\}}_{n=1}^{\infty }$ are periodic with period p  , for 88&_mathId=si13.gif&_user=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 88&_mathId=si14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">g_p$g_p$ between the metric subspaces 88&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_p$N_p$ and 88&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_p$V_p$, where 88&_mathId=si15.gif&_user=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 88&_mathId=si16.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=8838092044eda4abc463630ffd841d45" title="Click to view the MathML source">F_p$F_p$ of fixed points of 88&_mathId=si14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">g_p$g_p$ is exactly 88&_mathId=si17.gif&_user=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 88&_mathId=si18.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=aab7e30a4d21ea1cc37d71d6c8fcd01b" title="Click to view the MathML source">(p−1)$(p-1)$-dimensional submanifold of 88&_mathId=si19.gif&_user=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 88&_mathId=si10.gif&_user=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 88&_mathId=si15.gif&_user=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 88&_mathId=si10.gif&_user=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 88&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1">88-si11.gif">${\{c_n\}}_{n=1}^{\infty }$ and 88&_mathId=si12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8">88-si12.gif">${\{m_n\}}_{n=1}^{\infty }$ are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.