Density of algebras generated by z n and a smooth function on compact subsets of CC

In this note, we consider approximations of continuous functions on compact subsets K in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">C</m:mi> </m:math> C by elements of the algebra generated by z n and f , where f is a smooth function defined on some neighborhood of K . We wish to emphasize that the fact that the results presented herein constitute a natural extension of the chain of developments originating with Wermer’s work (1964) on uniform approximation on sufficiently small disks, now carried forward to uniform approximation on arbitrary polynomially convex compact subsets of the complex plane.