Wiener's Tauberian theorem

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 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 $G$, with $f \in L^1(G)$ and the Fourier transform 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 $0$.

Going deeper

For a student

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 $L^1(\R)$.