Faber polynomials

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)