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 i1" class="mathmlsrc">i1.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=c00fdb7b599c8631f42bd9b4f2084704">i1.gif">, with i101" class="mathmlsrc">i101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f">i101.gif"> 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 ${\{{\alpha }_n\}}_{n=0}^{\infty }$ are given by the relation<div class="formula" id="fm0010"><div class="mathml">${\alpha }_{n-1}={\bar{\rho }}_{n-1}\left[\frac{1-2m_n-i{c}_n}{1-i{c}_n}\right],n\ge 1,$div>div> where ρ₀=1${\rho }_0=1$, ${\rho }_n={\prod }_{k=1}^n(1-i{c}_k)/(1+i{c}_k)$, n≥1$n\ge 1$ and ${\{m_n\}}_{n=0}^{\infty }$ is the minimal parameter sequence of i101" class="mathmlsrc">i101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f">i101.gif">. In this paper we consider the space, denoted by i10" class="mathmlsrc">i10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_p, of all nontrivial probability measures such that the associated real sequences i11" class="mathmlsrc">i11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1">i11.gif"> and i12" class="mathmlsrc">i12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8">i12.gif"> are periodic with period p  , for i13" class="mathmlsrc">i13.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=38b82d41dd2132bfb22bea267bc698b3" title="Click to view the MathML source">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 i14" class="mathmlsrc">i14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">g_p between the metric subspaces i10" class="mathmlsrc">i10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_p and i15" class="mathmlsrc">i15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_p, where i15" class="mathmlsrc">i15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_p denotes the space of nontrivial probability measures with associated p  -periodic Verblunsky coefficients. Moreover, it is shown that the set i16" class="mathmlsrc">i16.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=8838092044eda4abc463630ffd841d45" title="Click to view the MathML source">F_p of fixed points of i14" class="mathmlsrc">i14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">g_p is exactly i17" class="mathmlsrc">i17.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=d00c29731cc554f4abc6356216db67a1" title="Click to view the MathML source">V_p∩N_p and this set is characterized by a i18" class="mathmlsrc">i18.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=aab7e30a4d21ea1cc37d71d6c8fcd01b" title="Click to view the MathML source">(p−1)-dimensional submanifold of i19" class="mathmlsrc">i19.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=af1316156a55e2691d427b2ce8c5a9e1" title="Click to view the MathML source">R^p. We also prove that the study of probability measures in i10" class="mathmlsrc">i10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">N_p is equivalent to the study of probability measures in i15" class="mathmlsrc">i15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">V_p. Furthermore, it is shown that the pure points of measures in i10" class="mathmlsrc">i10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">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 i11" class="mathmlsrc">i11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1">i11.gif"> and i12" class="mathmlsrc">i12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8">i12.gif"> are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.