Point Evaluations and Polynomial Approximation in the Mean

Abstract

We construct an example of a Borel subset E of the unit disk \(\{z : \mid z \mid < 1\}\) such that area measure restricted to E, which we denote AE, has the property that the set of bounded point evaluations for the polynomials with respect to the Lt(AE) norm varies with t. We further show that E can be chosen to be a simply connected region. In the context of smooth measures, like area measure, examples of this type were unexpected.