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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 L^1 or L^2 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 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 cannot 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 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 f is empty (in the case of L^1) 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 L^2). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L^1 group ring L^1(\mathbb) 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 ...
\mathbb 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 \mathbb. A similar result is true when \mathbb is replaced by any locally compact abelian group.


Introduction

A typical Tauberian theorem is the following result, for f\in L^1(0,\infty). If: # f(x)=O(1) as x\to\infty # \frac1x\int_0^\infty e^f(t)\,dt \to L as x\to\infty, then :\frac1x\int_0^xf(t)\,dt \to L. Generalizing, let G(t) be a given function, and P_G(f) be the proposition :\frac1x\int_0^\infty G(t/x)f(t)\,dt \to L. Note that one of the hypotheses and the conclusion of the Tauberian theorem has the form P_G(f), respectively, with G(t)=e^ and G(t)=1_(t). The second hypothesis is a "Tauberian condition". Wiener's Tauberian theorems have the following structure:, pp 385-377 :If G_1 is a given function such that W(G_1), P_(f), and R(f), then P_(f) holds for all "reasonable" G_2. Here R(f) is a "Tauberian" condition on f, and W(G_1) is a special condition on the kernel G_1. The power of the theorem is that P_(f) holds, not for a particular kernel G_2, but for ''all'' reasonable kernels G_2. The Wiener condition is roughly a condition on the zeros the Fourier transform of G_2. For instance, for functions of class L^1, 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 f\in L^1(\mathbb) be an integrable function. The span of translations f_a(x) = f(x+a) is dense in L^1(\mathbb) 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 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 f\in L^1 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 ...
f*h tends to zero at infinity for some h\in L^\infty. Then the convolution g*h tends to zero at infinity for any g\in L^1. More generally, if : \lim_ (f*h)(x) = A \int f(x) \,dx for some f\in L^1 the Fourier transform of which has no real zeros, then also : \lim_ (g*h)(x) = A \int g(x) \,dx for any g\in L^1.


Discrete version

Wiener's theorem has a counterpart in l^1(\mathbb): the span of the translations of f\in l^1(\mathbb) is dense if and only if the Fourier transform :\varphi(\theta) = \sum_ f(n) e^ \, has no real zeros. The following statements are equivalent version of this result: * Suppose the Fourier transform of f\in l^1(\mathbb) 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\in l^1(\mathbb). * Let \varphi 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 1/\varphi has absolutely convergent Fourier series if and only if \varphi has no zeros. showed that this is equivalent to the following property of the Wiener algebra A(\mathbb), 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 A(\mathbb) are all of the form ::M_x = \left\, \quad x \in \mathbb.


The condition in

Let f\in L^2(\mathbb) 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 f_a(x) = f(x+a) is dense in L^2(\mathbb) if and only if the real zeros of the Fourier transform of f form a set of zero Lebesgue measure. The parallel statement in l^2(\mathbb) is as follows: the span of translations of a sequence f\in l^2(\mathbb) is dense if and only if the zero set of the Fourier transform :\varphi(\theta) = \sum_ f(n) e^ 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