Bieberbach polynomials

Extremal polynomials which approximate a function that conformally maps a given simply-connected domain onto a disc. These polynomials were first studied by L. Bieberbach [1] in the context of the problem of the approximate computation of a conformal mapping.

Let $ G $ be a simply-connected domain in the finite part of the plane bounded by a curve $ \Gamma $, and let the function $ w = \phi (z) $ map this domain conformally and univalently onto the disc $ | w | < r _ {0} $ under the conditions $ \phi (z _ {0} ) = 0 $ and $ \phi ^ \prime (z _ {0} ) = 1 $, where $ z _ {0} $ is an arbitrary fixed point of $ G $ and $ r _ {0} $ depends on $ z _ {0} $. The polynomial $ \pi _ {n} (z) $ which minimizes the integral

$$ J (F _ {n} ) = \ {\int\limits \int\limits } _ {G} | F _ {n} ^ { \prime } (z) | ^ {2} dx dy $$

in the class of all polynomials $ F _ {n} (z) $ of degree $ n $ subject to the conditions $ F _ {n} (z _ {0} ) = 0 $ and $ F _ {n} ^ { \prime } (z _ {0} ) = 1 $ is called the Bieberbach polynomial. In the class of all functions which are analytic in the domain $ G $ and which satisfy the same conditions, this integral is minimized by the mapping function $ w = \phi (z) $. If the contour $ \Gamma $ is a Jordan curve, the sequence $ \{ \pi _ {n} (z) \} $ converges uniformly to the function $ \phi (z) $ inside $ G $. In the closed domain $ \overline{G}\; $ there need not be convergence [2]. If the contour $ \Gamma $ satisfies certain additional smoothness conditions, the sequence $ \{ \pi _ {n} (z) \} $ converges uniformly in the closed domain, and the rate of convergence depends on the degree of smoothness of $ \Gamma $.

References

[1]	L. Bieberbach, "Zur Theorie und Praxis der konformen Abbildung" Rend. Circ. Mat. Palermo , 38 (1914) pp. 98–112
[2]	M.V. Keldysh, "Sur l'approximation en moyenne quadratique des fonctions analytiques" Mat. Sb. , 5 (47) : 2 (1939) pp. 391–401
[3]	S.N. Mergelyan, "Some questions of the constructive theory of functions" Trudy Mat. Inst. Steklov. , 37 , Moscow (1951) (In Russian)
[4]	P.K. Suetin, "Polynomials orthogonal over a region and Bieberbach polynomials" Proc. Steklov Inst. Math. , 100 (1974) Trudy Mat. Inst. Steklov. , 100 (1971)

Comments

A good additional reference is [a1].

References

[a1]	D. Gaier, "Vorlesungen über Approximation im Komplexen" , Birkhäuser (1980)