Wiener's tauberian theorem

In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L₁ or L₂ can be approximated by linear combinations of translations of a given function.

Informally, if the Fourier transform of a function f vanishes on a certain set Z, the Fourier transform of any linear combination of translations of f also vanishes on Z. Therefore, the linear combinations of translations of f can not approximate a function whose Fourier transform does not vanish on Z.

Wiener's theorems make this precise, stating that linear combinations of translations of f are dense if and only the zero set of the Fourier transform of f is empty (in the case of L₁) or of Lebesgue measure zero (in the case of L₂).

Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L¹ group ring L¹(R) of the group R of real numbers is the dual group of R. A similar result is true when R is replaced by any locally compact abelian group.

The condition in L₁

Let f ∈ L₁(R) be an integrable function. The span of translations f_a(x) = f(x + a) is dense in L₁(R) if and only if the Fourier transform of f has no real zeros.

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 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

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

for some f ∈ L₁ the Fourier transform of which has no real zeros, then also

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

for any g ∈ L₁.

Discrete version

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

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

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 the convolution f * h tends to zero at infinity for some bounded sequence h. Then g * h 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

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

The condition in L₂

Let f ∈ L₂(R) be a square-integrable function. The span of translations f_a(x) = f(x + a) is dense in L₂(R) if and only if the real zeros of the Fourier transform of f form a set of zero Lebesgue measure.

The parallel statement in l₂(Z) is as follows: the span of translations of a sequence f ∈ l₂(Z) is dense if and only if the zero set of the Fourier transform

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

has zero Lebesgue measure.