50 Years with Hardy Spaces pp 173-190 | Cite as

# Bounded Point Derivations on Certain Function Algebras

## Abstract

Let *X* be a compact nowhere dense subset of the complex plane ℂ, let *C*(*X*) be the linear space of all continuous functions on *X* endowed with the uniform norm, and let *dA* denote two-dimensional Lebesgue (or area) measure in ℂ. Denote by *R*(*X*) the closure in *C*(*X*) of the set of all rational functions having no poles on *X*. It is well known that if *X* is sufficiently massive, then the functions in *R*(*X*) can inherit many of the properties usually associated with the analytic functions, such as unlimited degrees of differentiability and even the uniqueness property itself. Here we shall examine the extent to which some of those properties are inherited by the larger algebra *H ∞* (*X*), which by definition is the weak-*** closure of *R*(*X*) in *L*^{∞}(*X*) = *L*^{∞}(*X, dA*).

## Keywords

Point derivation monogeneity Swiss cheese peak point analytic capacity Wang’s theorem## Mathematics Subject Classification (2010)

Primary 30H50## Preview

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