Approximation by polynomials in $z$ and another function

Abstract:

We present some progress in the understanding of when and why polynomials in z and a given continuous function f are uniformly dense in all continuous complex valued functions on the closed unit disk. The first theorem requires that f be an ${\text {ACL}^2}$-function in a neighborhood of the disk which satisfies $\operatorname {Re} {f_{\bar z}} \geqslant |{f_z}|$ almost everywhere and ${f^{ - 1}}(f(a))$ is countable for each a. The second theorem requires that f have a special form and satisfy $|{f_{\bar z}}| > |{f_z}|$ everywhere except at the origin. The form is that $f = {\bar z^k}\phi (|z{|^{2k}})$ where $\phi$ is a complex valued function of a real variable satisfying $\phi$ is continuous in [0, 1], $\phi ’$ exists in (0, 1) and where k is a positive integer.