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) |
Faber polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Faber_polynomials&oldid=46897