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Wiener's Tauberian theorem

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In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L 1 {\displaystyle L^{1}} or L 2 {\displaystyle L^{2}} can be approximated by linear combinations of translations of a given function.

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

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

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

Introduction

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

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

then

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

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

1 x ∫ 0 ∞ G ( t / x ) f ( t ) d t → L . {\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 P G ( f ) {\displaystyle P_{G}(f)} , respectively, with G ( t ) = e − t {\displaystyle G(t)=e^{-t}} and G ( t ) = 1 [ 0 , 1 ] ( t ) . {\displaystyle G(t)=1_{}(t).} The second hypothesis is a "tauberian condition".

Wiener's tauberian theorems have the following structure:

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

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

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

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

More generally, if

lim x → ∞ ( f ∗ h ) ( x ) = A ∫ f ( x ) d x {\displaystyle \lim _{x\to \infty }(f*h)(x)=A\int f(x)\,dx}

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

lim x → ∞ ( g ∗ h ) ( x ) = A ∫ g ( x ) d x {\displaystyle \lim _{x\to \infty }(g*h)(x)=A\int g(x)\,dx}

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

Discrete version

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

φ ( θ ) = ∑ n ∈ Z f ( n ) e − i n θ {\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 f ∈ l 1 ( Z ) {\displaystyle f\in l^{1}(\mathbb {Z} )} has no real zeros, and for some bounded sequence h {\displaystyle h} the convolution f ∗ h {\displaystyle f*h}

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

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

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

Gelfand (1941a, 1941b) showed that this is equivalent to the following property of the Wiener algebra A ( T ) {\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 A ( T ) {\displaystyle A(\mathbb {T} )} are all of the form
M x = { f ∈ A ( T ) ∣ f ( x ) = 0 } , x ∈ T . {\displaystyle M_{x}=\left\{f\in A(\mathbb {T} )\mid f(x)=0\right\},\quad x\in \mathbb {T} .}

The condition in L

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

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

φ ( θ ) = ∑ n ∈ Z f ( n ) e − i n θ {\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

External links

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