---
tag: C96N6EB236
kind: definition
name: Wiener algebra
msc: [42A20, 46J10, 43A20]
defines: 5H8XESYCHQ
uses: [HHBGB5R5S2, 5NN2YZZR3V, 0X3WGX95FE, 9KYWN94QDN]
sources:
  - url: https://en.wikipedia.org/wiki/Wiener_algebra
---

## Definition

Let $\mathbb{T} = \R / 2\pi\Z$, and for an integrable function $f$ on $\mathbb{T}$ let $\hat f(n)$, $n \in \Z$, be its [[Fourier coefficients|5NN2YZZR3V]].
The *Wiener algebra* is the set
$$A(\mathbb{T}) = \Big\{ f \colon \mathbb{T} \to \C \text{ continuous} : \sum_{n \in \Z} \abs{\hat f(n)} < \infty \Big\},$$
of [[continuous|0X3WGX95FE]] functions with an [[absolutely convergent Fourier series|HHBGB5R5S2]], with pointwise addition and multiplication and the norm
$$\norm{f}_A = \sum_{n \in \Z} \abs{\hat f(n)}.$$

## 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|9KYWN94QDN]].
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|NC768TWZ3D]].