A QUANTITATIVE PROOF OF WIENER’S THEOREM

TERENCE TAO

Abstract. We give a quantitative proof of Wiener’s theorem based on the geometric series formula.

1. Introduction

Let l^{1}(Z) be the space of summable sequences (f(n))_{n∈Z} on the integers endowed with the norm ∥f∥_{l^{1}(Z)} := ∑_{n∈Z} |f(n)|. This space is spanned by the Kronecker deltas δ_{n}(m) is the Kronecker delta, defined as 1 when n = m and 0 when n ≠ m. We define convolution in the usual manner by specifying δ_{n} ∗ δ_{m} = δ_{n+m} and extending by bilinearity (so in particular δ_{0} is the convolution identity). From Young’s inequality we have

∥f ∗ g∥_{l^{1}(Z)} ≤ ∥f∥_{l^{1}(Z)}∥g∥_{l^{1}(Z)}.  (1)

The convolution structure can be converted to multiplication by the Fourier transform

f̂(e^{2πiθ}) := ∑_{n∈Z} f(n)e^{2πinθ} for all 0 ≤ θ ≤ 1  (2)

by the well-known formula { \atop f ∗ g} = f̂ĝ. In particular, if f, g ∈ l^{1}(Z) are such that f ∗ g = δ_{0}, then f̂(e^{2πiθ})ĝ(e^{2πiθ}) = 1 for all θ. In particular, f̂(e^{2πiθ}) is non-zero for all θ. A remarkable theorem of Wiener[4] asserts that the converse statement is also true:

Theorem 1.1. [4] Let f be an l^{1}(Z) function such that f̂(e^{2πiθ}) ≠ 0 for all 0 ≤ θ ≤ 1. Then there exists a g ∈ l^{1}(Z) such that f ∗ g = δ_{0}.

From Parseval’s identity it is easy enough to construct such a g in l^{2}(Z), and show that it is unique in that class; the challenge is to establish that the inverse g is in fact in l^{1}(Z).

We’ll give two proofs of this theorem. The first proof is short but nonquantitative; the second proof is an expansion of the first and is quantitative. Thus this note can be viewed as an example of how one to turn a non-quantitative argument into a quantitative one.

Brian Davies has pointed out that the arguments below are closely related to a short proof of Wiener’s theorem due to Newman [1]. Newman’s proof proceeds as below, but estimates ∥f∥_{l^{1}(Z)} in terms of the Lipschitz norm of f̂, which has the advantage of being easy to compute for repeated convolutions of f.

Date: January 30, 2020.

1991 Mathematics Subject Classification. 42A.

2. The non-quantitative proof.

Let I be the closed ideal in the convolution algebra l^{1}(Z) formed by the closure of f ∗l^{1}(Z), and let l^{1}(Z)/I be the quotient algebra. It will suffice to show that this quotient algebra is trivial, since this means that we can approximate the identity δ_{0} arbitrarily closely by functions of the form f ∗ g. But elements sufficiently close to δ_{0} are invertible by Neumann series, and hence f itself will be invertible as desired.

Suppose for contradiction that l^{1}(Z)/I is non-trivial. Let θ be arbitrary. Since f̂(e^{2πiθ}) is non-zero, f̂(e^{2πiθ})δ_{0} is invertible in l^{1}(Z), and hence f̂(e^{2πiθ})δ_{0} − f is invertible in l^{1}(Z)/I for all θ. But by (2)

f̂(e^{2πiθ})δ_{0} − f = ∑_{n∈Z} f(n)(e^{2πinθ}δ_{0} − δ_{n}).

Observe that each e^{2πinθ}δ_{0} − δ_{n} lies in the ideal generated by e^{2πiθ}δ_{0} − δ_{1}. Thus f̂(e^{2πiθ})δ_{0} − f lies in the closed ideal generated by e^{2πiθ}δ_{0} − δ_{1}. Since the former function was invertible in l^{1}(Z)/I, we thus see that the closed ideal generated by e^{2πiθ}δ_{0} − δ_{1} in l^{1}(Z)/I is in fact all of l^{1}(Z)/I. This is only possible if e^{2πiθ}δ_{0} − δ_{1} is invertible in l^{1}(Z)/I, by the Neumann series argument given earlier. In other words, δ_{1} has no spectrum on the unit circle. But as δ_{1} is unitary, it has no spectrum outside the unit circle either (again by Neumann series). Hence the resolvent operator (zδ_{0} − δ_{1})^{−1} is analytic and bounded on all of C, hence constant by Louville’s theorem. But this resolvent tends to 0 as z → ∞ and tends to −δ_{−1} as z → 0, hence −δ_{−1} = 0 in l^{1}(Z)/I. But −δ_{−1} is invertible, hence l^{1}(Z)/I is trivial as claimed.

3. The quantitative proof.

We now adapt the above argument to prove a more quantitative version of Theorem 1.1:

Theorem 3.1. Let f ∈ l^{1}(Z) be such that ∥f∥_{l^{1}(Z)} ≤ M for some 0 < M < ∞,

and |f̂(e^{2πiθ})| ≥ ε > 0 for all 0 ≤ θ ≤ 1 and some ε > 0. Suppose also that

∑_{n<n_{0} or n>n_{0}+N} |f(n)| ≤ ε/4  (3)

for some n_{0} ∈ Z and N ≥ 1. Then there exists a g ∈ l^{1}(Z) such that f ∗ g = δ_{0}, and furthermore

∥g∥_{l^{1}(Z)} ≤ 2048M^{2}N^{2}/ε^{3}.

Proof By multiplying f and M by 1/ε (and thus multiplying g by ε) we may reduce to the case ε = 1, which forces M ≥ 1. By translating f we may also take n_{0} = 0.

Let F ∈ l^{1}(Z) be the restriction of f to [0, N], thus F (n) := f(n) when 0 ≤ n ≤ N and F (n) := 0 when n > N or n < 0. Then by hypothesis

∥F − f∥_{l^{1}(Z)} ≤ 1/4  (4)

and hence by (2)

|F̂(e^{2πiθ}) − f̂(e^{2πiθ})| ≤ 1/4 for all θ.

By hypothesis, we have |f̂(e^{2πiθ})| ≥ 1, and hence

|F̂(e^{2πiθ})| ≥ 3/4 for all θ.  (5)

Since F has only finitely many non-zero values, we can extend the Fourier transform F̂ analytically to the entire complex plane as the polynomial

F̂(z) := ∑_{n∈Z} F (n)z^{n} = ∑_{n=0}^{N} f(n)z^{n}.

Observe that if z = re^{2πiθ} for some r > 0 with |r − 1| ≤ 1/8MN, then

|F̂(re^{iθ}) − F̂(e^{iθ})| ≤ ∑_{n=0}^{N} |f(n)||r^{n} − 1|

≤ ∑_{n=0}^{N} |f(n)|((1 + \frac{1}{8MN})^{N} − 1)

≤ ∥f∥_{l^{1}(Z)}((e^{1/8MN})^{N} − 1)

≤ M(e^{1/8M} − 1)

≤ e^{1/8} − 1

≤ 1/4,

where we have applied the fact that (e^{x} − 1)/x is a non-decreasing function of x for x > 0 to the values x = 1/8M and x = 1/8. From this and (5) we thus have

|F̂(z)| ≥ 1/2 for all 1 − 1/8MN ≤ |z| ≤ 1 + 1/8MN.

From this and (4) we can find a convolution inverse to h_{z} to the function F̂(z)δ_{0} − (F − f) in l^{1}(Z) by Neumann series, thus

(F̂(z)δ_{0} − (F − f)) ∗ h_{z} = δ_{0}.  (6)

Furthermore we have the Neumann series bound

∥h_{z}∥_{l^{1}} ≤ \frac{1}{|F̂(z)| − ∥F − f∥_{l^{1}(Z)}} ≤ \frac{1}{1/2 − 1/4} = 4.  (7)

Now observe the factorization

(F̂(z)δ_{0} − F) ∗ h_{z} = ∑_{n=0}^{N} f(n)(z^{n}δ_{0} − δ_{n}) ∗ h_{z} = (zδ_{0} − δ_{1}) ∗ H_{z}  (8)

where H_{z} is the function

H_{z} := [∑_{n=0}^{N} f(n)(z^{n−1}δ_{0} + z^{n−2}δ_{1} + … + zδ_{n−2} + δ_{n−1})] ∗ h_{z}.

Observe that H_{z} is bounded in l^{1}(Z), indeed by (1) and the triangle inequality (7) we have

\begin{aligned}∥H_{z}∥_{l^{1}(Z)} & ≤ ∑_{n=0}^{N} |f(n)|(|z|^{n−1} + … + 1)∥h_{z}∥_{l^{1}(Z)} \\ & ≤ 4 ∑_{n=0}^{N} |f(n)|N max(1, |z|)^{N} \\ & ≤ 4∥f∥_{l^{1}(Z)}N(1 + \frac{1}{8MN})^{N} \\ & ≤ 8MN \end{aligned}  (9)

(for instance). From (6), (8) we thus have

δ_{0} = (zδ_{0} − δ_{1}) ∗ H_{z} + f ∗ h_{z} for all 1 − \frac{1}{8MN} ≤ |z| ≤ 1 + \frac{1}{8MN} . (10)

Now we introduce the resolvents R_{z} for |z| ≠ 1 by the Neumann series

R_{z} := ∑_{k=0}^{∞} z^{−k−1}δ_{k} when |z| > 1  (11)

and

R_{z} := − ∑_{k=0}^{∞} δ_{−k−1}z^{k} when |z| < 1,  (12)

thus we have the Neumann series bounds

∥R_{z}∥_{l^{1}(Z)} ≤ \frac{1}{||z| − 1|} for all |z| ≠ 1.  (13)

and the resolvent identity

R_{z} ∗ (zδ_{0} − δ_{1}) = δ_{0} for all |z| ≠ 1.  (14)

If we set z := re^{2πiθ} and average (11), (12) over θ, we thus obtain

∫_{0}^{1} R_{re^{2πiθ}} dθ = 0 when r > 1

and

∫_{0}^{1} R_{re^{2πiθ}} dθ = −δ_{−1} when r < 1

(these can also be deduced from (14) and the Cauchy integral formula). In particular, if we set

σ := \frac{1}{128M^{2}N^{2}}  (15)

then we have the spectral resolution formula

δ_{−1} = ∫_{0}^{1}(R_{(1+σ)e^{2πiθ}} − R_{(1−σ)e^{2πiθ}}) dθ.  (16)

On the other hand, convolving (10) with R_{z} using (14), we have

R_{z} = H_{z} + f ∗ h_{z} ∗ R_{z} whenever 0 < ||z| − 1| ≤ 1/8MN.

The point is that, in contrast to (13), the bounds (9) on H_{z} do not blow up as |z| → 1. This discrepancy in behavior between R_{z} and H_{z} is what will eventually allow us to invert f.

Inserting the above identity into (16), we obtain

δ_{−1} = ∫_{0}^{1} H_{(1+σ)e^{2πiθ}} − H_{(1−σ)e^{2πiθ}} dθ + f ∗ G

where

G := ∫_{0}^{1}(h_{(1+σ)e^{2πiθ}} ∗ R_{(1+σ)e^{2πiθ}} − h_{(1−σ)e^{2πiθ}} ∗ R_{(1−σ)e^{2πiθ}}) dθ.  (17)

From Minkowski’s inequality and the fundamental theorem of calculus we thus see that

∥δ_{−1} − f ∗ G∥_{l^{1}(Z)} ≤ 2σ sup_{1−σ≤|z|≤1+σ} ∥\frac{d}{dz} H(z)∥_{l^{1}(Z)}.

On the other hand, then from (9) and the generalized Cauchy integral formula (applied to the circle of radius 1/16MN centered at z) we have

∥\frac{d}{dz} H(z)∥_{l^{1}(Z)} ≤ (16MN)(2MN).

By the above estimates and (15) we thus have

∥δ_{−1} − f ∗ G∥_{l^{1}(Z)} ≤ 64σM^{2}N^{2} = \frac{1}{2},

and hence by (1)

∥δ_{0} − f ∗ G ∗ δ_{1}∥_{l^{1}(Z)} ≤ \frac{1}{2}.

By Neumann series we may thus find a function J ∈ l^{1}(Z) with ∥J∥_{l^{1}(Z)} ≤ 2 such that

f ∗ G ∗ δ_{1} ∗ J = δ_{0},

and we have therefore successfully inverted f with g := G ∗ δ_{1} ∗ J. All that remains is to estimate the l^{1}(Z) norm of g. From (1), (17), Minkowski’s inequality, (7), (13), (15) we have

∥g∥_{l^{1}(Z)} ≤ 2∥G∥_{l^{1}(Z)}

≤ 4 sup_{||z|−1|=σ} ∥h_{z}∥_{l^{1}(Z)}∥R_{z}∥_{l^{1}(Z)}

≤ 16/σ

= 2048M^{2}N^{2}

as desired.

Remark 3.2. It is clear that Theorem 3.1 implies Theorem 1.1. Indeed, since f̂(e^{2πiθ}) is an absolutely summable Fourier series, it is periodic and continuous (by the Weierstrass M-test, or by dominated convergence), and thus if it is non-zero then it must be bounded from below in magnitude by some ε > 0. One can then locate n_{0}, N with the desired properties, again by the absolute convergence of f, although this integer N, while necessarily finite, can be arbitrarily large compared with M and ε.

One can ask if the dependence on this N parameter can be removed, which would lead to a much cleaner result. If M is very close to ε, for instance if M < \sqrt{2}ε, then the answer is yes. To see this, observe from Parseval’s identity that

ε^{2} ≤ ∫_{0}^{1} |f̂(e^{2πiθ})|^{2} dθ = ∑_{n∈Z} |f(n)|^{2} ≤ ∥f∥_{l^{1}(Z)} sup_{n} |f(n)| < \sqrt{2}ε sup_{n} |f(n)|

and hence there exists an n ∈ Z such that |f(n)| > ε/\sqrt{2} > M/2. Thus ∥f − f(n)δ_{n}∥_{l^{1}(Z)} = ∥f∥_{l^{1}(Z)} − |f(n)| < M/2, and f can then be inverted by Neumann series since f(n)δ_{n} is clearly invertible. However, this phenomenon breaks downwhen M and ε are further apart [3], [2]. Perhaps a more feasible conjecture is that a sharpening of Theorem 3.1 is possible in which the N large values of f are not assumed to be consecutive. For instance, instead of having the hypothesis (3), one might have

∑_{n∈Z:n≠n_{1},... ,n_{N}} |f(n)| ≤ ε/4

instead, where n_{1}, …, n_{N} are distinct integers (possibly very far apart), and the final bound still only depends on N, M, ε (i.e. it does not depend on the separation of the n_{j}).

I thank Anders Dahlner, Brian Davies, Thomas Strohmer and Vladimir Vassiliev for helpful comments.

References

[1] D.J. Newman, A simple proof of Wiener’s 1/f theorem, Proc. Amer. Math. Soc. 48 (1975), 264–265.

[2] H. S. Shapiro, A counterexample in harmonic analysis, Approximation theory (Papers, VIth Semester, Stefan Banach Internat. Math Center, Warsaw 1975), Banach Center Publ., vol 4, PWN, Warsaw 1979, 233–236.

[3] J.D. Stafney, An unbounded inverse property in the algebra of absolutely convergent Fourier series, Proc. Amer. Math. Soc. 18 (1967), 497–498.

[4] N. Wiener, Tauberian theorems, Ann. Math. 33 (1932), 1-100.

Department of Mathematics, UCLA, Los Angeles, CA 90024

   E-mail address: tao@math.ucla.edu