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 [math]\displaystyle{ L^1 }[/math] or [math]\displaystyle{ L^2 }[/math] can be approximated by linear combinations of translations of a given function.[2]
Informally, if the Fourier transform of a function [math]\displaystyle{ f }[/math] vanishes on a certain set [math]\displaystyle{ Z }[/math], the Fourier transform of any linear combination of translations of [math]\displaystyle{ f }[/math] also vanishes on [math]\displaystyle{ Z }[/math]. Therefore, the linear combinations of translations of [math]\displaystyle{ f }[/math] cannot approximate a function whose Fourier transform does not vanish on [math]\displaystyle{ Z }[/math].
Wiener's theorems make this precise, stating that linear combinations of translations of [math]\displaystyle{ f }[/math] are dense if and only if the zero set of the Fourier transform of [math]\displaystyle{ f }[/math] is empty (in the case of [math]\displaystyle{ L^1 }[/math]) or of Lebesgue measure zero (in the case of [math]\displaystyle{ L^2 }[/math]).
Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the [math]\displaystyle{ L^1 }[/math] group ring [math]\displaystyle{ L^1(\mathbb{R}) }[/math] of the group [math]\displaystyle{ \mathbb{R} }[/math] of real numbers is the dual group of [math]\displaystyle{ \mathbb{R} }[/math]. A similar result is true when [math]\displaystyle{ \mathbb{R} }[/math] is replaced by any locally compact abelian group.
The condition in L1
Let [math]\displaystyle{ f\in L^1(\mathbb{R}) }[/math] be an integrable function. The span of translations [math]\displaystyle{ f_a(x) = f(x+a) }[/math] is dense in [math]\displaystyle{ L^1(\mathbb{R}) }[/math] if and only if the Fourier transform of [math]\displaystyle{ f }[/math] has no real zeros.
Tauberian reformulation
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 [math]\displaystyle{ f\in L^1 }[/math] has no real zeros, and suppose the convolution [math]\displaystyle{ f*h }[/math] tends to zero at infinity for some [math]\displaystyle{ h\in L^\infty }[/math]. Then the convolution [math]\displaystyle{ g*h }[/math] tends to zero at infinity for any [math]\displaystyle{ g\in L^1 }[/math].
More generally, if
- [math]\displaystyle{ \lim_{x \to \infty} (f*h)(x) = A \int f(x) \,dx }[/math]
for some [math]\displaystyle{ f\in L^1 }[/math] the Fourier transform of which has no real zeros, then also
- [math]\displaystyle{ \lim_{x \to \infty} (g*h)(x) = A \int g(x) \,dx }[/math]
for any [math]\displaystyle{ g\in L^1 }[/math].
Discrete version
Wiener's theorem has a counterpart in [math]\displaystyle{ l^1(\mathbb{Z}) }[/math]: the span of the translations of [math]\displaystyle{ f\in l^1(\mathbb{Z}) }[/math] is dense if and only if the Fourier series
- [math]\displaystyle{ \varphi(\theta) = \sum_{n \in \mathbb{Z}} f(n) e^{-in\theta} \, }[/math]
has no real zeros. The following statements are equivalent version of this result:
- Suppose the Fourier series of [math]\displaystyle{ f\in l^1(\mathbb{Z}) }[/math] has no real zeros, and for some bounded sequence [math]\displaystyle{ h }[/math] the convolution [math]\displaystyle{ f*h }[/math]
tends to zero at infinity. Then [math]\displaystyle{ g*h }[/math] also tends to zero at infinity for any [math]\displaystyle{ g\in l^1(\mathbb{Z}) }[/math].
- Let [math]\displaystyle{ \varphi }[/math] be a function on the unit circle with absolutely convergent Fourier series. Then [math]\displaystyle{ 1/\varphi }[/math] has absolutely convergent Fourier series
if and only if [math]\displaystyle{ \varphi }[/math] has no zeros.
Gelfand (1941a, 1941b) showed that this is equivalent to the following property of the Wiener algebra [math]\displaystyle{ A(\mathbb{T}) }[/math], which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result:
- The maximal ideals of [math]\displaystyle{ A(\mathbb{T}) }[/math] are all of the form
- [math]\displaystyle{ M_x = \left\{ f \in A(\mathbb{T}) \mid f(x) = 0 \right\}, \quad x \in \mathbb{T}. }[/math]
The condition in L2
Let [math]\displaystyle{ f\in L^2(\mathbb{R}) }[/math] be a square-integrable function. The span of translations [math]\displaystyle{ f_a(x) = f(x+a) }[/math] is dense in [math]\displaystyle{ L^2(\mathbb{R}) }[/math] if and only if the real zeros of the Fourier transform of [math]\displaystyle{ f }[/math] form a set of zero Lebesgue measure.
The parallel statement in [math]\displaystyle{ l^2(\mathbb{Z}) }[/math] is as follows: the span of translations of a sequence [math]\displaystyle{ f\in l^2(\mathbb{Z}) }[/math] is dense if and only if the zero set of the Fourier series
- [math]\displaystyle{ \varphi(\theta) = \sum_{n \in \mathbb{Z}} f(n) e^{-in\theta} }[/math]
has zero Lebesgue measure.
Notes
References
- Gelfand, I. (1941a), "Normierte Ringe", Rec. Math. (Mat. Sbornik), Nouvelle Série 9 (51): 3–24
- Gelfand, I. (1941b), "Über absolut konvergente trigonometrische Reihen und Integrale", Rec. Math. (Mat. Sbornik), Nouvelle Série 9 (51): 51–66
- Rudin, W. (1991), Functional analysis, International Series in Pure and Applied Mathematics, New York: McGraw-Hill, Inc., ISBN 0-07-054236-8, https://archive.org/details/functionalanalys00rudi
- Wiener, N. (1932), "Tauberian Theorems", Annals of Mathematics 33 (1): 1–100, doi:10.2307/1968102
External links
- Hazewinkel, Michiel, ed. (2001), "Wiener Tauberian theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=W/w097950
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