It is known th
at given a pair of real sequences
class="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si1.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=c00fdb7b599c8631f42bd9b4f2084704"> class="imgLazyJSB inlineImage" height="17" width="139" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si1.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si1.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si1.gif" overflow="scroll">{ { c n } n = 1 ∞ , { d n } n = 1 ∞ } ath> , with
class="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f"> class="imgLazyJSB inlineImage" height="17" width="58" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si101.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si101.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si101.gif" overflow="scroll">{ d n } n = 1 ∞ ath> a positive chain sequence, we can associ
at e a unique nontrivial probability measure
μ on
the unit circle. Precisely,
the measure is such th
at the corresponding Verblunsky coefficients
class="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si3.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b36b296f671647cf0cfcf34f32ecb9fd"> class="imgLazyJSB inlineImage" height="17" width="60" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si3.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si3.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si3.gif" overflow="scroll">{ α n } n = 0 ∞ ath> are given by
the rel
at ion
class="formula" id="fm0010">
where
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si5.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=5b23b5ebcf9d4f3365601b2db8f1772d" title ="Click to view the Mat hML source">ρ0 =1 class="mat hContainer hidden">class="mat hCode">ath altimg="si5.gif" overflow="scroll">ρ 0 = 1 ath> ,
class="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si6.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=83d94d4f92f3e7f2d6cb8b526d62fa6d"> class="imgLazyJSB inlineImage" height="18" width="208" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si6.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si6.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si6.gif" overflow="scroll">ρ n = ∏ k = 1 n ( 1 − i c k ) / ( 1 + i c k ) ath> ,
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si490.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=76424479c0a197fe721e22aed714cdf2" title ="Click to view the Mat hML source">n≥1 class="mat hContainer hidden">class="mat hCode">ath altimg="si490.gif" overflow="scroll">n ≥ 1 ath> and
class="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si394.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=bb132d6a41ce02c224f0c250c51e6f61"> class="imgLazyJSB inlineImage" height="17" width="64" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si394.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si394.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si394.gif" overflow="scroll">{ m n } n = 0 ∞ ath> is
the minimal parameter sequence of
class="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f"> class="imgLazyJSB inlineImage" height="17" width="58" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si101.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si101.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si101.gif" overflow="scroll">{ d n } n = 1 ∞ ath> . In this paper we consider
the space, denoted by
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title ="Click to view the Mat hML source">Np class="mat hContainer hidden">class="mat hCode">ath altimg="si10.gif" overflow="scroll">N p ath> , of all nontrivial probability measures such th
at the associ
at ed real sequences
class="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1"> class="imgLazyJSB inlineImage" height="17" width="57" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si11.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si11.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si11.gif" overflow="scroll">{ c n } n = 1 ∞ ath> and
class="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8"> class="imgLazyJSB inlineImage" height="17" width="64" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si12.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si12.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si12.gif" overflow="scroll">{ m n } n = 1 ∞ ath> are periodic with period
p , for
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si13.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=38b82d41dd2132bfb22bea267bc698b3" title ="Click to view the Mat hML source">p∈N class="mat hContainer hidden">class="mat hCode">ath altimg="si13.gif" overflow="scroll">p ∈ athvariant="double-struck">N ath> . By assuming an appropri
at e metric on
the space of all nontrivial probability measures on
the unit circle, we show th
at the re exists a homeomorphism
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title ="Click to view the Mat hML source">gp class="mat hContainer hidden">class="mat hCode">ath altimg="si14.gif" overflow="scroll">g p ath> between
the metric subspaces
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title ="Click to view the Mat hML source">Np class="mat hContainer hidden">class="mat hCode">ath altimg="si10.gif" overflow="scroll">N p ath> and
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title ="Click to view the Mat hML source">Vp class="mat hContainer hidden">class="mat hCode">ath altimg="si15.gif" overflow="scroll">V p ath> , where
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title ="Click to view the Mat hML source">Vp class="mat hContainer hidden">class="mat hCode">ath altimg="si15.gif" overflow="scroll">V p ath> denotes
the space of nontrivial probability measures with associ
at ed
p -periodic Verblunsky coefficients. Moreover, it is shown th
at the set
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si16.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=8838092044eda4abc463630ffd841d45" title ="Click to view the Mat hML source">Fp class="mat hContainer hidden">class="mat hCode">ath altimg="si16.gif" overflow="scroll">F p ath> of fixed points of
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title ="Click to view the Mat hML source">gp class="mat hContainer hidden">class="mat hCode">ath altimg="si14.gif" overflow="scroll">g p ath> is exactly
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si17.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=d00c29731cc554f4abc6356216db67a1" title ="Click to view the Mat hML source">Vp ∩Np class="mat hContainer hidden">class="mat hCode">ath altimg="si17.gif" overflow="scroll">V p ∩ N p ath> and this set is characterized by a
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si18.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=aab7e30a4d21ea1cc37d71d6c8fcd01b" title ="Click to view the Mat hML source">(p−1) class="mat hContainer hidden">class="mat hCode">ath altimg="si18.gif" overflow="scroll">( p − 1 ) ath> -dimensional submanifold of
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si19.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=af1316156a55e2691d427b2ce8c5a9e1" title ="Click to view the Mat hML source">Rp class="mat hContainer hidden">class="mat hCode">ath altimg="si19.gif" overflow="scroll">athvariant="double-struck">R p ath> . We also prove th
at the study of probability measures in
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title ="Click to view the Mat hML source">Np class="mat hContainer hidden">class="mat hCode">ath altimg="si10.gif" overflow="scroll">N p ath> is equivalent to
the study of probability measures in
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title ="Click to view the Mat hML source">Vp class="mat hContainer hidden">class="mat hCode">ath altimg="si15.gif" overflow="scroll">V p ath> . Fur
the rmore, it is shown th
at the pure points of measures in
class="mat hmlsrc">class="formulat ext stixSupport mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title ="Click to view the Mat hML source">Np class="mat hContainer hidden">class="mat hCode">ath altimg="si10.gif" overflow="scroll">N p ath> are, in fact, zeros of associ
at ed 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="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1"> class="imgLazyJSB inlineImage" height="17" width="57" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si11.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si11.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si11.gif" overflow="scroll">{ c n } n = 1 ∞ ath> and
class="mat hmlsrc">title="View the Mat hML source" class ="mat hImg" dat a-mat hURL="/science?_ob=Mat hURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mat hId=si12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8"> class="imgLazyJSB inlineImage" height="17" width="64" alt="View the Mat hML source" title ="View the Mat hML source" src="/sd/grey_pxl.gif" dat a-inlimgeid="1-s2.0-S0022247X16304188-si12.gif"> the Mat hML source" title ="View the Mat hML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16304188-si12.gif"> class="mat hContainer hidden">class="mat hCode">ath altimg="si12.gif" overflow="scroll">{ m n } n = 1 ∞ ath> are limit periodic with period
p . Finally, we give some examples to illustr
at e
the results obtained.