Faber polynomials

A classical basis system that serves to represent analytic functions in a complex domain. Suppose that the complement of a bounded continuum containing more than one point is a simply-connected domain of the extended complex plane , and that the function , , is the conformal univalent mapping of onto the domain under the conditions and . Then the Faber polynomials can be defined as the sums of the terms of non-negative degree in in the Laurent expansions of the functions in a neighbourhood of the point . The Faber polynomials for can also be defined as the coefficients in the expansion

(1)

where the function is the inverse of . If is the disc , then . In the case when is the segment , the Faber polynomials are the Chebyshev polynomials of the first kind. These polynomials were introduced by G. Faber [1].

If is the closure of a simply-connected domain bounded by a rectifiable Jordan curve , and the function is analytic in , continuous in the closed domain and has bounded variation on , then it can be expanded in in a Faber series

(2)

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

The Faber series (2) converges uniformly in the closed domain if, for example, 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 , the Lebesgue inequality

holds for every function that is analytic in and continuous in , where the constant is independent of and , and is the best uniform approximation to in by polynomials of degree not exceeding .

One can introduce a weight function in the numerator of the left-hand side of (1), where is analytic in , is different from zero and . 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)