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 $f$ 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 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 $L^{1}$ group ring $L^{1}(\mathbb {R} )$ of the group $\mathbb {R}$ of real numbers is the dual group of $\mathbb {R}$ . A similar result is true when $\mathbb {R}$ is replaced by any locally compact abelian group.

The condition in $L 1$

Let $f\in L^{1}(\mathbb {R} )$ be an integrable function. The span of translations $f_{a}(x)=f(x+a)$ is dense in $L^{1}(\mathbb {R} )$ 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 $l^{1}(\mathbb {Z} )$ : the span of the translations of $f\in l^{1}(\mathbb {Z} )$ is dense if and only if the Fourier series

\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 series of $f\in l^{1}(\mathbb {Z} )$ 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 $g\in l^{1}(\mathbb {Z} )$ .

Let $\varphi$ be a function on the unit circle with absolutely convergent Fourier series. Then $1/\varphi$ has absolutely convergent Fourier series

if and only if $\varphi$ has no zeros.

Gelfand (1941a, 1941b) showed that this is equivalent to the following property of the Wiener algebra $A(\mathbb {T} )$ , which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result:

The maximal ideals of $A(\mathbb {T} )$ 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 $f\in L^{2}(\mathbb {R} )$ be a square-integrable function. The span of translations $f_{a}(x)=f(x+a)$ is dense in $L^{2}(\mathbb {R} )$ if and only if the real zeros of the Fourier transform of $f$ form a set of zero Lebesgue measure.

The parallel statement in $l^{2}(\mathbb {Z} )$ is as follows: the span of translations of a sequence $f\in l^{2}(\mathbb {Z} )$ is dense if and only if the zero set of the Fourier series

\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), Nouvelle Série, 9 (51): 3–24, MR 0004726
Gelfand, I. (1941b), "Über absolut konvergente trigonometrische Reihen und Integrale", Rec. Math. (Mat. Sbornik), Nouvelle Série, 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).