In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932.
[1] They provide a necessary and sufficient condition under which any function in
can be approximated by linear combinations of translations of a given function.
[2] Informally, if the Fourier transform of a function
, the Fourier transform of any linear combination of translations of
cannot approximate a function whose Fourier transform does not vanish on
Wiener's theorems make this precise, stating that linear combinations of translations of
are dense if and only if the zero set of the Fourier transform of
) or of Lebesgue measure zero (in the case of
Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the
of real numbers is the dual group of
is replaced by any locally compact abelian group.
A typical tauberian theorem is the following result, for
be the proposition Note that one of the hypotheses and the conclusion of the tauberian theorem has the form
The second hypothesis is a "tauberian condition".
Wiener's tauberian theorems have the following structure:[3] Here
, the condition is that the Fourier transform does not vanish anywhere.
The key point is that this easy necessary condition is also sufficient.
The following statement is equivalent to the previous result,[citation needed] and explains why Wiener's result is a Tauberian theorem: Suppose the Fourier transform of
has no real zeros, and suppose the convolution
the Fourier transform of which has no real zeros, then also for any
Wiener's theorem has a counterpart in
is dense if and only if the Fourier series has no real zeros.
The following statements are equivalent version of this result: tends to zero at infinity.
Gelfand (1941a, 1941b) showed that this is equivalent to the following property of the Wiener algebra
, which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result: Let
if and only if the real zeros of the Fourier transform of
form a set of zero Lebesgue measure.
is as follows: the span of translations of a sequence
is dense if and only if the zero set of the Fourier series has zero Lebesgue measure.