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 class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=c00fdb7b599c8631f42bd9b4f2084704">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">$\{{\{c_n\}}_{n=1}^{\infty },{\{d_n\}}_{n=1}^{\infty }\}$, with class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\{d_n\}}_{n=1}^{\infty }$ 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 class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b36b296f671647cf0cfcf34f32ecb9fd">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\{{\alpha }_n\}}_{n=0}^{\infty }$ are given by the relation where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=5b23b5ebcf9d4f3365601b2db8f1772d" title="Click to view the MathML source">ρ₀=1class="mathContainer hidden">class="mathCode">${\rho }_0=1$, class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=83d94d4f92f3e7f2d6cb8b526d62fa6d">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\rho }_n={\prod }_{k=1}^n(1-i{c}_k)/(1+i{c}_k)$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si490.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=76424479c0a197fe721e22aed714cdf2" title="Click to view the MathML source">n≥1class="mathContainer hidden">class="mathCode">$n\ge 1$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si394.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=bb132d6a41ce02c224f0c250c51e6f61">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\{m_n\}}_{n=0}^{\infty }$ is the minimal parameter sequence of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\{d_n\}}_{n=1}^{\infty }$. In this paper we consider the space, denoted by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_pclass="mathContainer hidden">class="mathCode">$N_p$, of all nontrivial probability measures such that the associated real sequences class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\{c_n\}}_{n=1}^{\infty }$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\{m_n\}}_{n=1}^{\infty }$ are periodic with period p  , for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si13.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=38b82d41dd2132bfb22bea267bc698b3" title="Click to view the MathML source">p∈Nclass="mathContainer hidden">class="mathCode">$p\in \mathbb{N}$. By assuming an appropriate metric on the space of all nontrivial probability measures on the unit circle, we show that there exists a homeomorphism class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">g_pclass="mathContainer hidden">class="mathCode">$g_p$ between the metric subspaces class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_pclass="mathContainer hidden">class="mathCode">$N_p$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_pclass="mathContainer hidden">class="mathCode">$V_p$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_pclass="mathContainer hidden">class="mathCode">$V_p$ denotes the space of nontrivial probability measures with associated p  -periodic Verblunsky coefficients. Moreover, it is shown that the set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si16.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=8838092044eda4abc463630ffd841d45" title="Click to view the MathML source">F_pclass="mathContainer hidden">class="mathCode">$F_p$ of fixed points of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">g_pclass="mathContainer hidden">class="mathCode">$g_p$ is exactly class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si17.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=d00c29731cc554f4abc6356216db67a1" title="Click to view the MathML source">V_p∩N_pclass="mathContainer hidden">class="mathCode">$V_p\cap N_p$ and this set is characterized by a class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si18.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=aab7e30a4d21ea1cc37d71d6c8fcd01b" title="Click to view the MathML source">(p−1)class="mathContainer hidden">class="mathCode">$(p-1)$-dimensional submanifold of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si19.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=af1316156a55e2691d427b2ce8c5a9e1" title="Click to view the MathML source">R^pclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^p$. We also prove that the study of probability measures in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_pclass="mathContainer hidden">class="mathCode">$N_p$ is equivalent to the study of probability measures in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_pclass="mathContainer hidden">class="mathCode">$V_p$. Furthermore, it is shown that the pure points of measures in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_pclass="mathContainer hidden">class="mathCode">$N_p$ 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 class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\{c_n\}}_{n=1}^{\infty }$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\{m_n\}}_{n=1}^{\infty }$ are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.