International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 3, Pages 415-426
doi:10.1155/S0161171279000326
Smoothness properties of functions in R2(x) at certain boundary points
Department of Mathematics, East Carolina University, Greenville 27834, North Carolina, USA
Received 31 January 1979
Copyright © 1979 Edwin Wolf. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let X be a compact subset of the complex plane ℂ. We denote by R0(X) the algebra consisting of the (restrictions to X of) rational functions with poles off X. Let m denote 2-dimensional Lebesgue measure. For p≥1, let Rp(X) be the closure of R0(X) in Lp(X,dm).
In this paper, we consider the case p=2. Let xϵ∂X be both a bounded point evaluation for R2(X) and the vertex of a sector contained in IntX. Let L be a line which passes through x and bisects the sector. For those yϵL∩X that are sufficiently near x we prove statements about |f(y)−f(x)| for all fϵR2(X).