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A classical basis system that serves to represent analytic functions in a complex domain. Suppose that the complement of a bounded continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f0380101.png" /> containing more than one point is a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f0380102.png" /> of the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f0380103.png" />, and that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f0380104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f0380105.png" />, is the conformal univalent mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f0380106.png" /> onto the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f0380107.png" /> under the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f0380108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f0380109.png" />. Then the Faber polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801010.png" /> can be defined as the sums of the terms of non-negative degree in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801011.png" /> in the Laurent expansions of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801012.png" /> in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801013.png" />. The Faber polynomials for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801014.png" /> can also be defined as the coefficients in the expansion
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<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/f/f038/f038010/f03801015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
  
where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801016.png" /> is the inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801018.png" /> is the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801020.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801021.png" /> is the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801022.png" />, the Faber polynomials are the [[Chebyshev polynomials|Chebyshev polynomials]] of the first kind. These polynomials were introduced by G. Faber [[#References|[1]]].
+
A classical basis system that serves to represent analytic functions in a complex domain. Suppose that the complement of a bounded continuum  $  K $
 +
containing more than one point is a simply-connected domain  $  D $
 +
of the extended complex plane  $  \mathbf C \cup \{ \infty \} $,
 +
and that the function $  w = \Phi ( z) $,
 +
$  z \in D $,
 +
is the conformal univalent mapping of $  D $
 +
onto the domain  $  | w | > 1 $
 +
under the conditions  $  \Phi ( \infty ) = \infty $
 +
and  $  \Phi  ^  \prime  ( \infty ) > 0 $.  
 +
Then the Faber polynomials $  \{ \Phi _ {n} ( z) \} $
 +
can be defined as the sums of the terms of non-negative degree in  $  z $
 +
in the Laurent expansions of the functions  $  \{ \Phi  ^ {n} ( z) \} $
 +
in a neighbourhood of the point  $  z = \infty $.  
 +
The Faber polynomials for  $  K $
 +
can also be defined as the coefficients in the expansion
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801023.png" /> is the closure of a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801024.png" /> bounded by a rectifiable Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801025.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801026.png" /> is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801027.png" />, continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801028.png" /> and has bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801029.png" />, then it can be expanded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801030.png" /> in a Faber series
+
$$ \tag{1 }
  
<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/f/f038/f038010/f03801031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{\Psi  ^  \prime  ( w) }{\Psi ( w) - z }
 +
  = \
 +
\sum _ {n = 0 } ^  \infty 
  
that converges uniformly inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801032.png" />, that is, on every closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801033.png" />, where the coefficients in the expansion are defined by the formula
+
\frac{\Phi _ {n} ( z) }{w ^ {n + 1 } }
 +
,\ \
 +
z \in K,\ \
 +
| w | > 1,
 +
$$
  
<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/f/f038/f038010/f03801034.png" /></td> </tr></table>
+
where the function  $  \zeta = \Psi ( w) $
 +
is the inverse of  $  w = \Phi ( \zeta ) $.
 +
If  $  K $
 +
is the disc  $  | z | \leq  1 $,
 +
then  $  \Phi _ {n} ( z) = z  ^ {n} $.
 +
In the case when  $  K $
 +
is the segment  $  [- 1, 1] $,
 +
the Faber polynomials are the [[Chebyshev polynomials|Chebyshev polynomials]] of the first kind. These polynomials were introduced by G. Faber [[#References|[1]]].
  
The Faber series (2) converges uniformly in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801035.png" /> if, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801036.png" /> has a continuously-turning tangent the angle of inclination to the real axis of which, as a function of the arc length, satisfies a Lipschitz condition. Under the same condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801037.png" />, the Lebesgue inequality
+
If  $  K $
 +
is the closure of a simply-connected domain  $  G $
 +
bounded by a rectifiable Jordan curve  $  \Gamma $,
 +
and the function  $  f ( z) $
 +
is analytic in  $  G $,
 +
continuous in the closed domain $  \overline{G}\; $
 +
and has bounded variation on  $  \Gamma $,  
 +
then it can be expanded in  $  G $
 +
in a Faber series
  
<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/f/f038/f038010/f03801038.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
f ( z)  = \
 +
\sum _ {n = 0 } ^  \infty 
 +
a _ {n} \Phi _ {n} ( z),\ \
 +
z \in G,
 +
$$
  
holds for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801039.png" /> that is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801040.png" /> and continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801041.png" />, where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801042.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801044.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801045.png" /> is the best uniform approximation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801047.png" /> by polynomials of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801048.png" />.
+
that converges uniformly inside  $  G $,
 +
that is, on every closed subset of $  G $,  
 +
where the coefficients in the expansion are defined by the formula
  
One can introduce a weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801049.png" /> in the numerator of the left-hand side of (1), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801050.png" /> is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801051.png" />, is different from zero and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038010/f03801052.png" />. Then the coefficients of the expansion (1) are called generalized Faber polynomials.
+
$$
 +
a _ {n}  = \
 +
{
 +
\frac{1}{2 \pi i }
 +
}
 +
\int\limits _  \Gamma
 +
 
 +
\frac{f ( \zeta ) \Phi  ^  \prime  ( \zeta ) }{\Phi ^ {n + 1 } ( \zeta ) }
 +
\
 +
d \zeta .
 +
$$
 +
 
 +
The Faber series (2) converges uniformly in the closed domain  $  \overline{G}\; $
 +
if, for example,  $  \Gamma $
 +
has a continuously-turning tangent the angle of inclination to the real axis of which, as a function of the arc length, satisfies a Lipschitz condition. Under the same condition on  $  \Gamma $,
 +
the Lebesgue inequality
 +
 
 +
$$
 +
\left | f ( z) -
 +
\sum _ {k = 0 } ^ { n }
 +
a _ {k} \Phi _ {k} ( z) \
 +
\right |  \leq  \
 +
c _ {1} E _ {n} ( f, \overline{G}\; ) \
 +
\mathop{\rm ln}  n,\ \
 +
z \in \overline{G}\; ,
 +
$$
 +
 
 +
holds for every function  $  f ( z) $
 +
that is analytic in  $  G $
 +
and continuous in  $  \overline{G}\; $,
 +
where the constant  $  c _ {1} $
 +
is independent of  $  n $
 +
and  $  z $,
 +
and  $  E _ {n} ( f, \overline{G}\; ) $
 +
is the best uniform approximation to  $  f ( z) $
 +
in  $  \overline{G}\; $
 +
by polynomials of degree not exceeding  $  n $.
 +
 
 +
One can introduce a weight function $  g [ \Psi ( w)] $
 +
in the numerator of the left-hand side of (1), where $  g ( z) $
 +
is analytic in $  D $,  
 +
is different from zero and $  g ( \infty ) > 0 $.  
 +
Then the coefficients of the expansion (1) are called generalized Faber polynomials.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Faber,  "Ueber polynomische Entwicklungen"  ''Math. Ann.'' , '''57'''  (1903)  pp. 389–408</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. Suetin,  "Series in Faber polynomials and several generalizations"  ''J. Soviet Math.'' , '''5'''  (1976)  pp. 502–551  ''Itogi Nauk. i Tekhn. Sovr. Probl. Mat.'' , '''5'''  (1975)  pp. 73–140</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.K. Suetin,  "Series in Faber polynomials" , Moscow  (1984)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Faber,  "Ueber polynomische Entwicklungen"  ''Math. Ann.'' , '''57'''  (1903)  pp. 389–408</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. Suetin,  "Series in Faber polynomials and several generalizations"  ''J. Soviet Math.'' , '''5'''  (1976)  pp. 502–551  ''Itogi Nauk. i Tekhn. Sovr. Probl. Mat.'' , '''5'''  (1975)  pp. 73–140</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.K. Suetin,  "Series in Faber polynomials" , Moscow  (1984)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:38, 5 June 2020


A classical basis system that serves to represent analytic functions in a complex domain. Suppose that the complement of a bounded continuum $ K $ containing more than one point is a simply-connected domain $ D $ of the extended complex plane $ \mathbf C \cup \{ \infty \} $, and that the function $ w = \Phi ( z) $, $ z \in D $, is the conformal univalent mapping of $ D $ onto the domain $ | w | > 1 $ under the conditions $ \Phi ( \infty ) = \infty $ and $ \Phi ^ \prime ( \infty ) > 0 $. Then the Faber polynomials $ \{ \Phi _ {n} ( z) \} $ can be defined as the sums of the terms of non-negative degree in $ z $ in the Laurent expansions of the functions $ \{ \Phi ^ {n} ( z) \} $ in a neighbourhood of the point $ z = \infty $. The Faber polynomials for $ K $ can also be defined as the coefficients in the expansion

$$ \tag{1 } \frac{\Psi ^ \prime ( w) }{\Psi ( w) - z } = \ \sum _ {n = 0 } ^ \infty \frac{\Phi _ {n} ( z) }{w ^ {n + 1 } } ,\ \ z \in K,\ \ | w | > 1, $$

where the function $ \zeta = \Psi ( w) $ is the inverse of $ w = \Phi ( \zeta ) $. If $ K $ is the disc $ | z | \leq 1 $, then $ \Phi _ {n} ( z) = z ^ {n} $. In the case when $ K $ is the segment $ [- 1, 1] $, the Faber polynomials are the Chebyshev polynomials of the first kind. These polynomials were introduced by G. Faber [1].

If $ K $ is the closure of a simply-connected domain $ G $ bounded by a rectifiable Jordan curve $ \Gamma $, and the function $ f ( z) $ is analytic in $ G $, continuous in the closed domain $ \overline{G}\; $ and has bounded variation on $ \Gamma $, then it can be expanded in $ G $ in a Faber series

$$ \tag{2 } f ( z) = \ \sum _ {n = 0 } ^ \infty a _ {n} \Phi _ {n} ( z),\ \ z \in G, $$

that converges uniformly inside $ G $, that is, on every closed subset of $ G $, where the coefficients in the expansion are defined by the formula

$$ a _ {n} = \ { \frac{1}{2 \pi i } } \int\limits _ \Gamma \frac{f ( \zeta ) \Phi ^ \prime ( \zeta ) }{\Phi ^ {n + 1 } ( \zeta ) } \ d \zeta . $$

The Faber series (2) converges uniformly in the closed domain $ \overline{G}\; $ if, for example, $ \Gamma $ has a continuously-turning tangent the angle of inclination to the real axis of which, as a function of the arc length, satisfies a Lipschitz condition. Under the same condition on $ \Gamma $, the Lebesgue inequality

$$ \left | f ( z) - \sum _ {k = 0 } ^ { n } a _ {k} \Phi _ {k} ( z) \ \right | \leq \ c _ {1} E _ {n} ( f, \overline{G}\; ) \ \mathop{\rm ln} n,\ \ z \in \overline{G}\; , $$

holds for every function $ f ( z) $ that is analytic in $ G $ and continuous in $ \overline{G}\; $, where the constant $ c _ {1} $ is independent of $ n $ and $ z $, and $ E _ {n} ( f, \overline{G}\; ) $ is the best uniform approximation to $ f ( z) $ in $ \overline{G}\; $ by polynomials of degree not exceeding $ n $.

One can introduce a weight function $ g [ \Psi ( w)] $ in the numerator of the left-hand side of (1), where $ g ( z) $ is analytic in $ D $, is different from zero and $ g ( \infty ) > 0 $. Then the coefficients of the expansion (1) are called generalized Faber polynomials.

References

[1] G. Faber, "Ueber polynomische Entwicklungen" Math. Ann. , 57 (1903) pp. 389–408
[2] P.K. Suetin, "Series in Faber polynomials and several generalizations" J. Soviet Math. , 5 (1976) pp. 502–551 Itogi Nauk. i Tekhn. Sovr. Probl. Mat. , 5 (1975) pp. 73–140
[3] P.K. Suetin, "Series in Faber polynomials" , Moscow (1984) (In Russian)

Comments

[a1] is a general reference concerning approximation of functions of a complex variable. It contains a section on Faber expansions.

References

[a1] D. Gaier, "Vorlesungen über Approximation im Komplexen" , Birkhäuser (1980)
[a2] J.H. Curtiss, "Faber polynomials and Faber series" Amer. Math. Monthly , 78 (1971) pp. 577–596
[a3] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 3.14 (Translated from Russian)
How to Cite This Entry:
Faber polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Faber_polynomials&oldid=17377
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article