Holomorphic functional calculus in the Wiener algebra

Let $\mathrm{\Omega }$ be an open set on which $\mathrm{\Phi }$ is holomorphic and which contains $f(\mathbb{T})$. The compact-neighbourhood lemma A compact subset of an open Euclidean set has a compact Jordan neighborhood inside that open set gives a finite union $J$ of closed grid rectangles with $f(\mathbb{T})\subseteq \mathrm{int}J\subseteq J\subseteq \mathrm{\Omega }$. Orient the frontier edges of the constituent grid cells positively and cancel each internal edge against its reverse. The resulting polygonal chain $\mathrm{\Gamma }$ is a cycle in $\mathrm{\Omega }\setminus f(\mathbb{T})$, with $n(\mathrm{\Gamma },w)=1$ for $w\in f(\mathbb{T})$ and $n(\mathrm{\Gamma },w)=0$ for $w\mathrm{\notin }\mathrm{\Omega }$: summing the cell indices first proves this away from the grid lines, and local constancy of the cycle index extends it to every point off the frontier. Thus $\mathrm{\Gamma }$ is null-homologous in $\mathrm{\Omega }$. For $z\in {\mathrm{\Gamma }}^{\ast }$, $z-f$ has no zero on $\mathbb{T}$, so $(z-f{)}^{-1}\in A(\mathbb{T})$ by [L1].

The map $z\mapsto \mathrm{\Phi }(z)(z-f{)}^{-1}$ is continuous into $A(\mathbb{T})$ (the inverse identity follows from $(z-f{)}^{-1}-(w-f{)}^{-1}=(w-z)(z-f{)}^{-1}(w-f{)}^{-1}$). Hence its normalized chain integral $F:=\frac{1}{2\pi i}{\int }_{\mathrm{\Gamma }}\mathrm{\Phi }(z)(z-f{)}^{-1}dz$ is an $A(\mathbb{T})$ element, as the finite sum of norm-limits of edgewise Riemann sums by [L2].

Evaluation at $x$ commutes with those norm-limits. Since step 1.1 gives $n(\mathrm{\Gamma },f(x))=1$ and makes $\mathrm{\Gamma }$ null-homologous in $\mathrm{\Omega }$, [L3] applied to $\mathrm{\Phi }$ gives $F(x)=\mathrm{\Phi }(f(x))$. Thus $F=\mathrm{\Phi }\circ f$ and belongs to $A(\mathbb{T})$.