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)