International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 3, Pages 459-466
doi:10.1155/S016117128300040X
Some remarks on the space R2(E)
Department of Mathematics, University of Stockholm, Sweden
Received 26 April 1982; Revised 28 September 1982
Copyright © 1983 Claes Fernström. 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 E be a compact subset of the complex plane. We denote by R(E) the algebra consisting of the rational functions with poles off E. The closure of R(E) in Lp(E), 1≤p<∞, is denoted by Rp(E). In this paper we consider the case p=2. In section 2 we introduce the notion of weak bounded point evaluation of order β and identify the existence of a weak bounded point evaluation of order β, β>1, as a necessary and sufficient condition for R2(E)≠L2(E). We also construct a compact set E such that R2(E) has an isolated bounded point evaluation. In section 3 we examine the smoothness properties of functions in R2(E) at those points which admit bounded point evaluations.