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Extremal polynomials which approximate a function that conformally maps a given simply-connected domain onto a disc. These polynomials were first studied by L. Bieberbach [[#References|[1]]] in the context of the problem of the approximate computation of a conformal mapping.
 
Extremal polynomials which approximate a function that conformally maps a given simply-connected domain onto a disc. These polynomials were first studied by L. Bieberbach [[#References|[1]]] in the context of the problem of the approximate computation of a conformal mapping.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b0161701.png" /> be a simply-connected domain in the finite part of the plane bounded by a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b0161702.png" />, and let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b0161703.png" /> map this domain conformally and univalently onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b0161704.png" /> under the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b0161705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b0161706.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b0161707.png" /> is an arbitrary fixed point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b0161708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b0161709.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617010.png" />. The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617011.png" /> which minimizes the integral
+
Let $  G $
 +
be a simply-connected domain in the finite part of the plane bounded by a curve $  \Gamma $,  
 +
and let the function $  w = \phi (z) $
 +
map this domain conformally and univalently onto the disc $  | w | < r _ {0} $
 +
under the conditions $  \phi (z _ {0} ) = 0 $
 +
and $  \phi  ^  \prime  (z _ {0} ) = 1 $,  
 +
where $  z _ {0} $
 +
is an arbitrary fixed point of $  G $
 +
and $  r _ {0} $
 +
depends on $  z _ {0} $.  
 +
The polynomial $  \pi _ {n} (z) $
 +
which minimizes the integral
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617012.png" /></td> </tr></table>
+
$$
 +
J (F _ {n} )  = \
 +
{\int\limits \int\limits } _ {G} | F _ {n} ^ { \prime } (z) |  ^ {2}  dx  dy
 +
$$
  
in the class of all polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617013.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617014.png" /> subject to the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617016.png" /> is called the Bieberbach polynomial. In the class of all functions which are analytic in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617017.png" /> and which satisfy the same conditions, this integral is minimized by the mapping function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617018.png" />. If the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617019.png" /> is a Jordan curve, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617020.png" /> converges uniformly to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617021.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617022.png" />. In the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617023.png" /> there need not be convergence [[#References|[2]]]. If the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617024.png" /> satisfies certain additional smoothness conditions, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617025.png" /> converges uniformly in the closed domain, and the rate of convergence depends on the degree of smoothness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016170/b01617026.png" />.
+
in the class of all polynomials $  F _ {n} (z) $
 +
of degree $  n $
 +
subject to the conditions $  F _ {n} (z _ {0} ) = 0 $
 +
and $  F _ {n} ^ { \prime } (z _ {0} ) = 1 $
 +
is called the Bieberbach polynomial. In the class of all functions which are analytic in the domain $  G $
 +
and which satisfy the same conditions, this integral is minimized by the mapping function $  w = \phi (z) $.  
 +
If the contour $  \Gamma $
 +
is a Jordan curve, the sequence $  \{ \pi _ {n} (z) \} $
 +
converges uniformly to the function $  \phi (z) $
 +
inside $  G $.  
 +
In the closed domain $  \overline{G}\; $
 +
there need not be convergence [[#References|[2]]]. If the contour $  \Gamma $
 +
satisfies certain additional smoothness conditions, the sequence $  \{ \pi _ {n} (z) \} $
 +
converges uniformly in the closed domain, and the rate of convergence depends on the degree of smoothness of $  \Gamma $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,  "Zur Theorie und Praxis der konformen Abbildung"  ''Rend. Circ. Mat. Palermo'' , '''38'''  (1914)  pp. 98–112</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.V. Keldysh,  "Sur l'approximation en moyenne quadratique des fonctions analytiques"  ''Mat. Sb.'' , '''5 (47)''' :  2  (1939)  pp. 391–401</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.N. Mergelyan,  "Some questions of the constructive theory of functions"  ''Trudy Mat. Inst. Steklov.'' , '''37''' , Moscow  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.K. Suetin,  "Polynomials orthogonal over a region and Bieberbach polynomials"  ''Proc. Steklov Inst. Math.'' , '''100'''  (1974)  ''Trudy Mat. Inst. Steklov.'' , '''100'''  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,  "Zur Theorie und Praxis der konformen Abbildung"  ''Rend. Circ. Mat. Palermo'' , '''38'''  (1914)  pp. 98–112</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.V. Keldysh,  "Sur l'approximation en moyenne quadratique des fonctions analytiques"  ''Mat. Sb.'' , '''5 (47)''' :  2  (1939)  pp. 391–401</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.N. Mergelyan,  "Some questions of the constructive theory of functions"  ''Trudy Mat. Inst. Steklov.'' , '''37''' , Moscow  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.K. Suetin,  "Polynomials orthogonal over a region and Bieberbach polynomials"  ''Proc. Steklov Inst. Math.'' , '''100'''  (1974)  ''Trudy Mat. Inst. Steklov.'' , '''100'''  (1971)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 10:59, 29 May 2020


Extremal polynomials which approximate a function that conformally maps a given simply-connected domain onto a disc. These polynomials were first studied by L. Bieberbach [1] in the context of the problem of the approximate computation of a conformal mapping.

Let $ G $ be a simply-connected domain in the finite part of the plane bounded by a curve $ \Gamma $, and let the function $ w = \phi (z) $ map this domain conformally and univalently onto the disc $ | w | < r _ {0} $ under the conditions $ \phi (z _ {0} ) = 0 $ and $ \phi ^ \prime (z _ {0} ) = 1 $, where $ z _ {0} $ is an arbitrary fixed point of $ G $ and $ r _ {0} $ depends on $ z _ {0} $. The polynomial $ \pi _ {n} (z) $ which minimizes the integral

$$ J (F _ {n} ) = \ {\int\limits \int\limits } _ {G} | F _ {n} ^ { \prime } (z) | ^ {2} dx dy $$

in the class of all polynomials $ F _ {n} (z) $ of degree $ n $ subject to the conditions $ F _ {n} (z _ {0} ) = 0 $ and $ F _ {n} ^ { \prime } (z _ {0} ) = 1 $ is called the Bieberbach polynomial. In the class of all functions which are analytic in the domain $ G $ and which satisfy the same conditions, this integral is minimized by the mapping function $ w = \phi (z) $. If the contour $ \Gamma $ is a Jordan curve, the sequence $ \{ \pi _ {n} (z) \} $ converges uniformly to the function $ \phi (z) $ inside $ G $. In the closed domain $ \overline{G}\; $ there need not be convergence [2]. If the contour $ \Gamma $ satisfies certain additional smoothness conditions, the sequence $ \{ \pi _ {n} (z) \} $ converges uniformly in the closed domain, and the rate of convergence depends on the degree of smoothness of $ \Gamma $.

References

[1] L. Bieberbach, "Zur Theorie und Praxis der konformen Abbildung" Rend. Circ. Mat. Palermo , 38 (1914) pp. 98–112
[2] M.V. Keldysh, "Sur l'approximation en moyenne quadratique des fonctions analytiques" Mat. Sb. , 5 (47) : 2 (1939) pp. 391–401
[3] S.N. Mergelyan, "Some questions of the constructive theory of functions" Trudy Mat. Inst. Steklov. , 37 , Moscow (1951) (In Russian)
[4] P.K. Suetin, "Polynomials orthogonal over a region and Bieberbach polynomials" Proc. Steklov Inst. Math. , 100 (1974) Trudy Mat. Inst. Steklov. , 100 (1971)

Comments

A good additional reference is [a1].

References

[a1] D. Gaier, "Vorlesungen über Approximation im Komplexen" , Birkhäuser (1980)
How to Cite This Entry:
Bieberbach polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bieberbach_polynomials&oldid=15907
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article