In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, Wiener's Tauberian theorem is any of several related results
proved by
Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American computer scientist, mathematician, and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology ( MIT). A child prodigy, Wiener late ...
in 1932. They provide a necessary and sufficient condition under which any
function in
or
can be approximated by
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
s of
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
of a given function.
Informally, if the
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a function
vanishes on a certain set
, the Fourier transform of any linear combination of translations of
also vanishes on
. Therefore, the linear combinations 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
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the
zero set
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or eq ...
of the Fourier
transform of
is
empty (in the case of
) or of
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
zero (in the case of
).
Gelfand reformulated Wiener's theorem in terms of
commutative C*-algebras, when it states that the spectrum of the
group ring
of the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s is the dual group of
. A similar result is true when
is replaced by any
locally compact abelian group.
Introduction
A typical Tauberian theorem is the following result, for
. If:
#
as
#
as
,
then
:
Generalizing, let
be a given function, and
be the proposition
:
Note that one of the hypotheses and the conclusion of the Tauberian theorem has the form
, respectively, with
and
The second hypothesis is a "Tauberian condition".
Wiener's Tauberian theorems have the following structure:
[, pp 385-377]
:If
is a given function such that
,
, and
, then
holds for all "reasonable"
.
Here
is a "Tauberian" condition on
, and
is a special condition on the kernel
. The power of the theorem is that
holds, not for a particular kernel
, but for ''all'' reasonable kernels
.
The Wiener condition is roughly a condition on the zeros the Fourier transform of
. For instance, for functions of class
, the condition is that the Fourier transform does not vanish anywhere. This condition is often easily seen to be a ''necessary'' condition for a Tauberian theorem of this kind to hold. The key point is that this easy
necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
is also sufficient.
The condition in
Let
be an
integrable function. The
span of translations
is dense in
if and only if the Fourier transform of
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
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing tha ...
:
Suppose the Fourier transform of
has no real zeros, and suppose the
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
tends to zero at infinity for some
. Then the convolution
tends to zero at infinity for any
.
More generally, if
:
for some
the Fourier transform of which has no real zeros, then also
:
for any
.
Discrete version
Wiener's theorem has a counterpart in
: the span of the translations of
is dense if and only if the Fourier transform
:
has no real zeros. The following statements are equivalent version of this result:
* Suppose the Fourier transform of
has no real zeros, and for some bounded sequence
the convolution
tends to zero at infinity. Then
also tends to zero at infinity for any
.
* Let
be a function on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
with
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
. Then
has absolutely convergent Fourier series
if and only if
has no zeros.
showed that this is equivalent to the following property of the
Wiener algebra ,
which he proved using the theory of
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
s, thereby giving a new proof of Wiener's result:
* The
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
s of
are all of the form
::
The condition in
Let
be a
square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
. The span of translations
is dense in
if and only if the real zeros of the Fourier transform of
form a set of
zero Lebesgue measure.
The parallel statement in
is as follows: the span of translations of a sequence
is dense if and only if the zero set of the Fourier transform
:
has zero Lebesgue measure.
Notes
References
*
*
*
*
External links
*{{eom, id=W/w097950, title=Wiener Tauberian theorem, first=A.I., last=Shtern
Real analysis
Harmonic analysis
Tauberian theorems