Tauberian reformulation

The following statement is equivalent to the previous result, and explains why Wiener's result is a Tauberian theorem:

Suppose the Fourier transform of ${\displaystyle f\in L^{1}}$ has no real zeros, and suppose the convolution ${\displaystyle f*h}$ tends to zero at infinity for some ${\displaystyle h\in L^{\infty }}$. Then the convolution ${\displaystyle g*h}$ tends to zero at infinity for any ${\displaystyle g\in L^{1}}$.

More generally, if

${\displaystyle \lim _{x\to \infty }(f*h)(x)=A\int f(x)\,dx}$

for some ${\displaystyle f\in L^{1}}$ the Fourier transform of which has no real zeros, then also

${\displaystyle \lim _{x\to \infty }(g*h)(x)=A\int g(x)\,dx}$

for any ${\displaystyle g\in L^{1}}$.

Discrete version

Wiener's theorem has a counterpart in ${\displaystyle l^{1}(\mathbb {Z} )}$: the span of the translations of ${\displaystyle f\in l^{1}(\mathbb {Z} )}$ is dense if and only if the Fourier series

${\displaystyle \varphi (\theta )=\sum _{n\in \mathbb {Z} }f(n)e^{-in\theta }\,}$

has no real zeros. The following statements are equivalent version of this result:

• Suppose the Fourier series of ${\displaystyle f\in l^{1}(\mathbb {Z} )}$ has no real zeros, and for some bounded sequence ${\displaystyle h}$ the convolution ${\displaystyle f*h}$

tends to zero at infinity. Then ${\displaystyle g*h}$ also tends to zero at infinity for any ${\displaystyle g\in l^{1}(\mathbb {Z} )}$.

• Let ${\displaystyle \varphi }$ be a function on the unit circle with absolutely convergent Fourier series. Then ${\displaystyle 1/\varphi }$ has absolutely convergent Fourier series

if and only if ${\displaystyle \varphi }$ has no zeros.

Gelfand (1941a, 1941b) showed that this is equivalent to the following property of the Wiener algebra ${\displaystyle A(\mathbb {T} )}$, which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result:

• The maximal ideals of ${\displaystyle A(\mathbb {T} )}$ are all of the form

${\displaystyle M_{x}=\left\{f\in A(\mathbb {T} )\mid f(x)=0\right\},\quad x\in \mathbb {T} .}$