Wiener's Tauberian theorem

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 $L 1$ or $L 2$ can be approximated by linear combinations of translations of a given function.[2]

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 if the zero set of the Fourier transform of $f$ is empty (in the case of $L 1$ ) or of Lebesgue measure 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 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 $L 1$

Let $f \in L 1 (R)$ be an integrable function. The span of translations $f a (x)$ = $f (x + a)$ is dense in $L 1 (R)$ 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 $f \in L 1$ has no real zeros, and suppose the convolution $f * h$ 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 _{x\to \infty }(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 _{x\to \infty }(g*h)(x)=A\int g(x)\,dx

for any $g \in L 1$ .

Discrete version

Wiener's theorem has a counterpart in $l 1 (Z)$ : the span of the translations of $f \in l 1 (Z)$ 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 $f \in l 1 (Z)$ 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 $g \in l 1 (Z)$ .
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 $L 2$

Let $f \in L 2 (R)$ be a square-integrable function. The span of translations $f a (x)$ = $f (x + a)$ is dense in $L 2 (R)$ if and only if the real zeros of the Fourier transform of $f$ form a set of zero Lebesgue measure.

The parallel statement in $l 2 (Z)$ is as follows: the span of translations of a sequence $f \in l 2 (Z)$ 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.

Notes

See Wiener (1932).
see Rudin (1991).

References

Gelfand, I. (1941a), "Normierte Ringe", Rec. Math. (Mat. Sbornik) N.S., 9 (51): 3–24, MR 0004726
Gelfand, I. (1941b), "Über absolut konvergente trigonometrische Reihen und Integrale", Rec. Math. (Mat. Sbornik) N.S., 9 (51): 51–66, MR 0004727
Rudin, W. (1991), Functional analysis, International Series in Pure and Applied Mathematics, New York: McGraw-Hill, Inc., ISBN 0-07-054236-8, MR 1157815
Wiener, N. (1932), "Tauberian Theorems", Annals of Mathematics, 33 (1): 1–100, doi:10.2307/1968102, JSTOR 1968102

External links

Shtern, A.I. (2001) [1994], "Wiener Tauberian theorem", Encyclopedia of Mathematics, EMS Press

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[1] See Wiener (1932).

[2] see Rudin (1991).