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| In [[complex analysis]], the '''Blaschke product''' is a bounded [[analytic function]] in the open unit disc constructed to have zeros at a (finite or infinite) sequence of prescribed [[complex number]]s
| | 28 yr old Clothing Patternmaker Sandy Gohr from Iqaluit, loves to spend some time magic, diet and antiques. Felt particulary stimulated after touring Historic Centre of Sighisoara. |
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| :''a''<sub>0</sub>, ''a''<sub>1</sub>, ...
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| inside the [[unit disc]].
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| Blaschke products were introduced by {{harvs|txt|authorlink=Wilhelm Blaschke|first=Wilhelm |last=Blaschke|year=1915}}. They are related to [[Hardy space]]s.
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| ==Definition==
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| A sequence of points <math>(a_n)</math> inside the unit disk is said to satisfy the '''Blaschke condition''' when
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| <!--
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| :Σ (1 − |''a''<sub>''n''</sub>|)
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| :<math>\sum_n (1-|a_n|) <\infty.</math>
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| Given a sequence obeying the Blaschke condition, the Blaschke product is defined as
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| <!--
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| :''B''(''z'') = Π ''B''(''a''<sub>''n''</sub>, ''z'')
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| :<math>B(z)=\prod_n B(a_n,z)</math>
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| with factors
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| <!--
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| :''B''(''a'',''z'') = |''a''|/''a''·(''z'' − ''a'')/(1 − ''a''<sup>*</sup>''z'')
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| -->
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| :<math>B(a,z)=\frac{|a|}{a}\;\frac{a-z}{1 - \overline{a}z}</math>
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| provided ''a'' ≠ 0. Here <math>\overline{a}</math> is the [[complex conjugate]] of ''a''. When ''a'' = 0 take ''B''(''0'',''z'') = ''z''.
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| The Blaschke product ''B''(''z'') defines a function analytic in the open unit disc, and zero exactly at the ''a''<sub>''n''</sub> (with [[Multiplicity (mathematics)|multiplicity]] counted): furthermore it is in the Hardy class <math>H^\infty</math>.<ref>Conway (1996) 274</ref>
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| The sequence of ''a''<sub>''n''</sub> satisfying the convergence criterion above is sometimes called a '''Blaschke sequence'''.
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| ==Szegő theorem==
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| A theorem of [[Gábor Szegő]] states that if ''f'' is in <math>H^1</math>, the [[Hardy space]] with integrable norm, and if ''f'' is not identically zero, then the zeroes of ''f'' (certainly countable in number) satisfy the Blaschke condition.
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| ==Finite Blaschke products==
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| Finite Blaschke products can be characterized (as analytic functions on the unit disc) in the following way: Assume that ''f'' is an analytic function on the open unit disc such that
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| ''f'' can be extended to a continuous function on the closed unit disc
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| : <math>\overline{\Delta}= \{z \in \mathbb{C}\,|\, |z|\le 1\} </math>
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| which maps the unit circle to itself. Then ƒ is equal to a finite Blaschke product
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| :<math> B(z)=\zeta\prod_{i=1}^n\left({{z-a_i}\over {1-\overline{a_i}z}}\right)^{m_i}
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| </math>
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| where ''ζ'' lies on the unit circle and ''m<sub>i</sub>'' is the [[Multiplicity (mathematics)|multiplicity]] of the zero ''a<sub>i</sub>'', |''a''<sub>''i''</sub>| < 1. In particular, if ''ƒ'' satisfies the condition above and has no zeros inside the unit circle then ''ƒ'' is constant (this fact is also a consequence of the [[maximum principle]] for [[harmonic function]]s, applied to the harmonic function log(|''ƒ''(''z'')|)).
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| ==See also==
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| * [[Hardy space]]
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| * [[Weierstrass product]]
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| ==References==
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| {{reflist}}
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| * W. Blaschke, ''Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen'' Berichte Math.-Phys. Kl., Sächs. Gesell. der Wiss. Leipzig, 67 (1915) pp. 194–200
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| * Peter Colwell, ''Blaschke Products — Bounded Analytic Functions'' (1985), University of Michigan Press, Ann Arbor. ISBN 0-472-10065-3
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| * {{cite book | title=Functions of a Complex Variable II | volume=159 | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | isbn=0-387-94460-5 | first=John B. | last=Conway | authorlink=John B. Conway | pages=273–274 }}
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| *{{springer|id=b/b016630|first=P.M.|last= Tamrazov}}
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| [[Category:Complex analysis]]
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28 yr old Clothing Patternmaker Sandy Gohr from Iqaluit, loves to spend some time magic, diet and antiques. Felt particulary stimulated after touring Historic Centre of Sighisoara.