Example

Let A be the complex Banach algebra of functions on the unit circle with absolutely convergent Fourier series (I, p. 19, Example 8). Let $\varphi \in$ A. Let $f$ be a germ of a holomorphic function in a neighbourhood of the set of values of $\varphi$. Then $\psi = \Theta_{\varphi}(f)$ is an absolutely convergent Fourier series which for every $u\in \mathbf{U}$ satisfies $\psi (u) =f(\varphi (u))$ (Corollary 1, applied to the characters $\varphi \mapsto \varphi (u)$). In other words, the function $f\circ \varphi$ on the unit circle also has an absolutely convergent Fourier series (“theorem of P. Lévy”). This result generalises Wiener's theorem (I, p. 38, Example 4), which concerns the case of the function $f(z) = 1/z$ on $\mathbf{C}-\{0\}$ when $\varphi$ does not vanish.