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 ge" height="17" width="139" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si1.gif">$\{{\{c_n\}}_{n=1}^{\infty },{\{d_n\}}_{n=1}^{\infty }\}$, with ge" height="17" width="58" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-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 ge" height="17" width="60" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si3.gif">${\{{\alpha }_n\}}_{n=0}^{\infty }$ are given by the relation where ρ₀=1${\rho }_0=1$, ge" height="18" width="208" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si6.gif">${\rho }_n={\prod }_{k=1}^n(1-i{c}_k)/(1+i{c}_k)$, n&ge;1$n\&<font\; color="red">ge</font>;1$ and ge" height="17" width="64" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si394.gif">${\{m_n\}}_{n=0}^{\infty }$ is the minimal parameter sequence of ge" height="17" width="58" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si101.gif">${\{d_n\}}_{n=1}^{\infty }$. In this paper we consider the space, denoted by N_p$N_p$, of all nontrivial probability measures such that the associated real sequences ge" height="17" width="57" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si11.gif">${\{c_n\}}_{n=1}^{\infty }$ and ge" height="17" width="64" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si12.gif">${\{m_n\}}_{n=1}^{\infty }$ are periodic with period p  , for 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 g_p$g_p$ between the metric subspaces N_p$N_p$ and V_p$V_p$, where V_p$V_p$ denotes the space of nontrivial probability measures with associated p  -periodic Verblunsky coefficients. Moreover, it is shown that the set F_p$F_p$ of fixed points of g_p$g_p$ is exactly V_p∩N_p$V_p\cap N_p$ and this set is characterized by a (p−1)$(p-1)$-dimensional submanifold of R^p${\mathbb{R}}^p$. We also prove that the study of probability measures in N_p$N_p$ is equivalent to the study of probability measures in V_p$V_p$. Furthermore, it is shown that the pure points of measures in 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 ge" height="17" width="57" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si11.gif">${\{c_n\}}_{n=1}^{\infty }$ and ge" height="17" width="64" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si12.gif">${\{m_n\}}_{n=1}^{\infty }$ are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.