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 be a simply-connected domain in the finite part of the plane bounded by a curve
, and let the function
map this domain conformally and univalently onto the disc
under the conditions
and
, where
is an arbitrary fixed point of
and
depends on
. The polynomial
which minimizes the integral
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in the class of all polynomials of degree
subject to the conditions
and
is called the Bieberbach polynomial. In the class of all functions which are analytic in the domain
and which satisfy the same conditions, this integral is minimized by the mapping function
. If the contour
is a Jordan curve, the sequence
converges uniformly to the function
inside
. In the closed domain
there need not be convergence [2]. If the contour
satisfies certain additional smoothness conditions, the sequence
converges uniformly in the closed domain, and the rate of convergence depends on the degree of smoothness of
.
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) |
Bieberbach polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bieberbach_polynomials&oldid=15907