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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  $  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|Chebyshev polynomials]] of the first kind. These polynomials were introduced by G. Faber [[#References|[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====
<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====
[[#References|[a1]]] is a general reference concerning [[Approximation of functions of a complex variable|approximation of functions of a complex variable]]. It contains a section on Faber expansions.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Gaier,   "Vorlesungen über Approximation im Komplexen" , Birkhäuser  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Curtiss,   "Faber polynomials and Faber series"  ''Amer. Math. Monthly'' , '''78'''  (1971)  pp. 577–596</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Chapt. 3.14  (Translated from Russian)</TD></TR></table>