Best M\"obius approximations of convex and concave mappings
Martin Chuaqui, Brad Osgood
TL;DR
This paper investigates optimal M"obius approximations for convex and concave conformal mappings of the disk, emphasizing the importance of pole placement and leveraging properties of Blaschke products for polygonal mappings.
Contribution
It introduces a detailed analysis of the best M"obius approximations for convex and concave mappings, highlighting the role of pole locations and special properties for polygonal cases.
Findings
Pole location critically influences approximation quality
Blaschke product properties aid in polygon mapping approximations
Finer approximation details are achievable for polygonal mappings
Abstract
We study the best M\"obius approximations (BMA) to convex and concave conformal mappings of the disk, including the special case of mappings onto convex polygons. The crucial factor is the location of the poles of the BMAs. Finer details are possible in the case of polygons through special properties of Blaschke products and the prevertices of the mapping function.
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Taxonomy
TopicsAnalytic and geometric function theory