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

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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 src">Img" 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"><img 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">

img="si1.gif" overflow="scroll"> {{c_{n}}_{n = 1}^{\infty}, {d_{n}}_{n = 1}^{\infty}}

, with src"><img 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">

img="si101.gif" overflow="scroll"> {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 src"><img 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">

img="si3.gif" overflow="scroll"> {α_{n}}_{n = 0}^{\infty}

are given by the relation

src"><img class="imgLazyJSB inlineImage" height="40" width="284" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si4.gif">

img="si4.gif" overflow="scroll"> α_{n - 1} = {\overline{ρ}}_{n - 1} [\frac{1 - 2 m_{n} - i c_{n}}{1 - i c_{n}}], n \geq 1,

<img class="temp" src="/sd/blank.gif">

where src">ρ₀=1

img="si5.gif" overflow="scroll"> ρ_{0} = 1

, src"><img 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">

img="si6.gif" overflow="scroll"> ρ_{n} = \prod_{k = 1}^{n} (1 - i c_{k}) / (1 + i c_{k})

, src">n≥1

img="si490.gif" overflow="scroll"> n \geq 1

and src"><img 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">

img="si394.gif" overflow="scroll"> {m_{n}}_{n = 0}^{\infty}

is the minimal parameter sequence of src"><img 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">

img="si101.gif" overflow="scroll"> {d_{n}}_{n = 1}^{\infty}

. In this paper we consider the space, denoted by src">N_p

img="si10.gif" overflow="scroll"> N_{p}

, of all nontrivial probability measures such that the associated real sequences src"><img 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">

img="si11.gif" overflow="scroll"> {c_{n}}_{n = 1}^{\infty}

and src"><img 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">

img="si12.gif" overflow="scroll"> {m_{n}}_{n = 1}^{\infty}

are periodic with period p , for src">p∈N

img="si13.gif" overflow="scroll"> p \in 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 src">g_p

img="si14.gif" overflow="scroll"> g_{p}

between the metric subspaces src">N_p

img="si10.gif" overflow="scroll"> N_{p}

and src">V_p

img="si15.gif" overflow="scroll"> V_{p}

, where src">V_p

img="si15.gif" overflow="scroll"> V_{p}

denotes the space of nontrivial probability measures with associated p -periodic Verblunsky coefficients. Moreover, it is shown that the set src">F_p

img="si16.gif" overflow="scroll"> F_{p}

of fixed points of src">g_p

img="si14.gif" overflow="scroll"> g_{p}

is exactly src">V_p∩N_p

img="si17.gif" overflow="scroll"> V_{p} \cap N_{p}

and this set is characterized by a src">(p−1)

img="si18.gif" overflow="scroll"> (p - 1)

-dimensional submanifold of src">R^p

img="si19.gif" overflow="scroll"> R^{p}

. We also prove that the study of probability measures in src">N_p

img="si10.gif" overflow="scroll"> N_{p}

is equivalent to the study of probability measures in src">V_p

img="si15.gif" overflow="scroll"> V_{p}

. Furthermore, it is shown that the pure points of measures in src">N_p

img="si10.gif" overflow="scroll"> 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 src"><img 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">

img="si11.gif" overflow="scroll"> {c_{n}}_{n = 1}^{\infty}

and src"><img 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">

img="si12.gif" overflow="scroll"> {m_{n}}_{n = 1}^{\infty}

are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.

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