---
tag: V20376HXHD
kind: statement
role: theorem
name: Wiener's Tauberian theorem
msc: [43A20, "40E05", 42A38]
uses: [DPVM34ADZV, ZR09S9R3AW, VHN5VW5Q92]
about: [DPVM34ADZV]
sources:
  - url: https://en.wikipedia.org/wiki/Wiener%27s_Tauberian_theorem
same_as:
  wikidata: Q3527279
  wikipedia: Wiener's Tauberian theorem
---

## Statement

Let $f \in L^1(\R)$, the Lebesgue space of integrable functions, and let $\widehat{f}(\xi) = \int_\R f(x) e^{-2\pi i x\xi} \, dx$ be its [[Fourier transform|DPVM34ADZV]].
For $a \in \R$ let $f_a(x) = f(x - a)$.
Then the linear span of the translates $\set{f_a : a \in \R}$ is dense in $L^1(\R)$ if and only if $\widehat{f}(\xi) \ne 0$ for every $\xi \in \R$.

## Remarks

The "only if" part is easy: if $\widehat{f}(\xi_0) = 0$, then every $g$ in the closed span has $\widehat{g}(\xi_0) = 0$, because $\widehat{f_a}(\xi) = e^{-2\pi i a\xi}\widehat{f}(\xi)$ and $g \mapsto \widehat{g}(\xi_0)$ is continuous on $L^1$; but some $g \in L^1$ has $\widehat{g}(\xi_0) \ne 0$.
The "if" part uses [[Wiener's lemma|NC768TWZ3D]] type arguments: closed translation-invariant subspaces of $L^1(\R)$ are ideals for convolution, and an ideal whose transforms have no common zero is dense.
Example: the Gaussian $f(x) = e^{-\pi x^2}$ has $\widehat{f}(\xi) = e^{-\pi\xi^2} > 0$, so linear combinations of its translates approximate every integrable function in $L^1$.
The same statement holds for every [[locally compact abelian group|ZR09S9R3AW]] $G$, with $f \in L^1(G)$ and the [[Fourier transform|VHN5VW5Q92]] on the dual group.
There is also an $L^2$ version: the translates of $f \in L^2(\R)$ span a dense subspace of $L^2(\R)$ if and only if the zero set of $\widehat{f}$ has Lebesgue measure $0$.

## Explanations

### L1 human

Take one integrable function and all its shifted copies.
The theorem tells when sums of these copies can approximate every integrable function.
The answer is: exactly when the function contains every frequency, that is, its Fourier transform never vanishes.

### L2 human

Wiener used the theorem to give general Tauberian theorems: if $\phi$ is bounded and $f * \phi(x) \to c \int f$ as $x \to \infty$ for one $f$ with nonvanishing transform, then the same holds for all $g \in L^1$.
This gives a short route to Tauberian results for summation methods and, through the Wiener–Ikehara theorem, to the prime number theorem.
In Banach algebra language, the theorem says that a closed ideal of $L^1(\R)$ whose Fourier transforms have no common zero is all of $L^1(\R)$.