Extremal quasiconformality vs rational approximation
Samuel L. Krushkal
TL;DR
This paper investigates the limitations of weighted bounded rational approximation in hyperbolic domains, revealing that extremal quasiconformality imposes significant restrictions on the approximation of most holomorphic functions.
Contribution
It establishes a connection between extremal quasiconformality and the scarcity of rational approximations in hyperbolic domains, highlighting new obstructions to approximation.
Findings
Weighted bounded rational approximation is only possible for a sparse set of functions.
Obstructions are caused by features of extremal quasiconformality.
Most holomorphic functions cannot be approximated in the weighted bounded rational sense.
Abstract
We show that on most of the hyperbolic simply connected domains the weighted bounded rational approximation in a natural sup norm is possible only for a very sparse set of holomorphic functions (in contrast to integral approximation). The obstructions are caused by the features of extremal quasiconformality.
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