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 ${\displaystyle L^{1}}$ or ${\displaystyle L^{2}}$ can be approximated by linear combinations of translations of a given function. ^[2]

Informally, if the Fourier transform of a function ${\displaystyle f}$ vanishes on a certain set ${\displaystyle Z}$, the Fourier transform of any linear combination of translations of ${\displaystyle f}$ also vanishes on ${\displaystyle Z}$. Therefore, the linear combinations of translations of ${\displaystyle f}$ cannot approximate a function whose Fourier transform does not vanish on ${\displaystyle Z}$.

Wiener's theorems make this precise, stating that linear combinations of translations of ${\displaystyle f}$ are dense if and only if the zero set of the Fourier transform of ${\displaystyle f}$ is empty (in the case of ${\displaystyle L^{1}}$) or of Lebesgue measure zero (in the case of ${\displaystyle L^{2}}$).

Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the ${\displaystyle L^{1}}$ group ring ${\displaystyle L^{1}(\mathbb {R} )}$ of the group ${\displaystyle \mathbb {R} }$ of real numbers is the dual group of ${\displaystyle \mathbb {R} }$. A similar result is true when ${\displaystyle \mathbb {R} }$ is replaced by any locally compact abelian group.

Introduction

A typical tauberian theorem is the following result, for ${\displaystyle f\in L^{1}(0,\infty )}$. If:

1. ${\displaystyle f(x)=O(1)}$ as ${\displaystyle x\to \infty }$
2. ${\displaystyle {\frac {1}{x}}\int _{0}^{\infty }e^{-t/x}f(t)\,dt\to L}$ as ${\displaystyle x\to \infty }$,

then

${\displaystyle {\frac {1}{x}}\int _{0}^{x}f(t)\,dt\to L.}$

Generalizing, let ${\displaystyle G(t)}$ be a given function, and ${\displaystyle P_{G}(f)}$ be the proposition

${\displaystyle {\frac {1}{x}}\int _{0}^{\infty }G(t/x)f(t)\,dt\to L.}$

Note that one of the hypotheses and the conclusion of the tauberian theorem has the form ${\displaystyle P_{G}(f)}$, respectively, with ${\displaystyle G(t)=e^{-t}}$ and ${\displaystyle G(t)=1_{[0,1]}(t).}$ The second hypothesis is a "tauberian condition".

Wiener's tauberian theorems have the following structure: ^[3]

If ${\displaystyle G_{1}}$ is a given function such that ${\displaystyle W(G_{1})}$, ${\displaystyle P_{G_{1}}(f)}$, and ${\displaystyle R(f)}$, then ${\displaystyle P_{G_{2}}(f)}$ holds for all "reasonable" ${\displaystyle G_{2}}$.

Here ${\displaystyle R(f)}$ is a "tauberian" condition on ${\displaystyle f}$, and ${\displaystyle W(G_{1})}$ is a special condition on the kernel ${\displaystyle G_{1}}$. The power of the theorem is that ${\displaystyle P_{G_{2}}(f)}$ holds, not for a particular kernel ${\displaystyle G_{2}}$, but for all reasonable kernels ${\displaystyle G_{2}}$.

The Wiener condition is roughly a condition on the zeros the Fourier transform of ${\displaystyle G_{2}}$. For instance, for functions of class ${\displaystyle L^{1}}$, 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 is also sufficient.

The condition in L¹

Let ${\displaystyle f\in L^{1}(\mathbb {R} )}$ be an integrable function. The span of translations ${\displaystyle f_{a}(x)=f(x+a)}$ is dense in ${\displaystyle L^{1}(\mathbb {R} )}$ if and only if the Fourier transform of ${\displaystyle f}$ 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 ${\displaystyle f\in L^{1}}$ has no real zeros, and suppose the convolution ${\displaystyle f*h}$ tends to zero at infinity for some ${\displaystyle h\in L^{\infty }}$. Then the convolution ${\displaystyle g*h}$ tends to zero at infinity for any ${\displaystyle g\in L^{1}}$.

More generally, if

${\displaystyle \lim _{x\to \infty }(f*h)(x)=A\int f(x)\,dx}$

for some ${\displaystyle f\in L^{1}}$ the Fourier transform of which has no real zeros, then also

${\displaystyle \lim _{x\to \infty }(g*h)(x)=A\int g(x)\,dx}$

for any ${\displaystyle g\in L^{1}}$.

Discrete version

Wiener's theorem has a counterpart in ${\displaystyle l^{1}(\mathbb {Z} )}$: the span of the translations of ${\displaystyle f\in l^{1}(\mathbb {Z} )}$ is dense if and only if the Fourier series

${\displaystyle \varphi (\theta )=\sum _{n\in \mathbb {Z} }f(n)e^{-in\theta }\,}$

has no real zeros. The following statements are equivalent version of this result:

• Suppose the Fourier series of ${\displaystyle f\in l^{1}(\mathbb {Z} )}$ has no real zeros, and for some bounded sequence ${\displaystyle h}$ the convolution ${\displaystyle f*h}$

tends to zero at infinity. Then ${\displaystyle g*h}$ also tends to zero at infinity for any ${\displaystyle g\in l^{1}(\mathbb {Z} )}$.

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

if and only if ${\displaystyle \varphi }$ has no zeros.

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

• The maximal ideals of ${\displaystyle A(\mathbb {T} )}$ are all of the form

${\displaystyle M_{x}=\left\{f\in A(\mathbb {T} )\mid f(x)=0\right\},\quad x\in \mathbb {T} .}$

The condition in L²

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

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

${\displaystyle \varphi (\theta )=\sum _{n\in \mathbb {Z} }f(n)e^{-in\theta }}$

has zero Lebesgue measure.

Notes

1. See Wiener (1932).
2. see Rudin (1991).
3. G H Hardy, Divergent series, pp 385-377

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