Wiener algebra

Remarks

The map $f \mapsto (\hat f(n))_{n \in \Z}$ is an isometric isomorphism of $A(\mathbb{T})$ onto $\ell^1(\Z)$. It turns pointwise products into convolutions: $\widehat{fg}(n) = \sum_{k \in \Z} \hat f(k) \hat g(n - k)$. So $\norm{fg}_A \le \norm{f}_A \norm{g}_A$, and $A(\mathbb{T})$ is a commutative unital Banach algebra. Every $f \in A(\mathbb{T})$ satisfies $\sup_x \abs{f(x)} \le \norm{f}_A$.

Example: $f(x) = 2 + \cos x$ lies in $A(\mathbb{T})$ with $\hat f(0) = 2$, $\hat f(\pm 1) = 1/2$, and $\norm{f}_A = 3$.

Non-example: $f(x) = \sum_{n \ge 2} \frac{\sin nx}{n \log n}$. Its coefficients decrease to $0$ and $n \cdot \frac{1}{n \log n} \to 0$, so by the Chaundy–Jolliffe theorem the series converges uniformly and $f$ is continuous. But $\abs{\hat f(\pm n)} = \frac{1}{2 n \log n}$ and $\sum_{n \ge 2} \frac{1}{n \log n} = \infty$, so $f \notin A(\mathbb{T})$.

The characters of $A(\mathbb{T})$ are the evaluations $f \mapsto f(x)$ at points $x \in \mathbb{T}$; this is the key to Wiener's lemma.