Wiener's Tauberian theorem

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

Suppose the Fourier transform of f ∈ L¹ has no real zeros, and suppose the convolution f * h tends to zero at infinity for some h ∈ L^∞. Then the convolution g * h tends to zero at infinity for any g ∈ L¹.

More generally, if

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

for some f ∈ L¹ 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 g ∈ L¹.

Wiener's theorem has a counterpart in l¹(Z): the span of the translations of f ∈ l¹(Z) is dense if and only if the Fourier transform

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

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

• Suppose the Fourier transform of f ∈ l¹(Z) has no real zeros, and for some bounded sequence h the convolution f * h tends to zero at infinity. Then g * h also tends to zero at infinity for any g ∈ l¹(Z).
• Let φ be a function on the unit circle with absolutely convergent Fourier series. Then 1/φ has absolutely convergent Fourier series if and only if φ has no zeros.

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

• The maximal ideals of A(T) are all of the form

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