 Analytic Wiener algebra

The analytic Wiener algebra consists of the power series $f(z)={\sum }_{n\ge 0}{a}_n{z}^n$ with ${\sum }_{n\ge 0}|a_n|<\infty$, identified with their values on the unit circle. The norm is $\Vert f\Vert =\sum |a_n|$. Discrete convolution gives multiplication and a submultiplicative norm; completeness follows from the absolutely summable sequence space. Each series converges uniformly on the closed unit disk and is holomorphic in its interior.