$L$-analytic mappings in the disk algebra

Abstract:

It is shown that two classes of function transformations coincide when the transformations take place within the disk algebra. The first class is that of the $L$-analytic mappings. These are the ones given locally by power series: $f \to {\sum {{g_n}(f - {f_0})} ^n}$. The second class is that of locally pointwise mappings. A mapping $f \to \Phi [f]$ is pointwise if it has the form $\Phi [f](x) = {\Phi ^\ast }(x,f(x))$. It is a by-product of the disk algebra investigation that if a set $X$ has certain topological properties, then every locally pointwise mapping in $C(X)$ is continuous.