Wiener's lemma

In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the circle to its atomic part. This result admits an analogous statement for measures on the real line. It was first discovered by Norbert Wiener.

Statement

• Given a real or complex Borel measure [math]\displaystyle{ \mu }[/math] on the unit circle [math]\displaystyle{ \mathbb T }[/math], let [math]\displaystyle{ \mu_a=\sum_j c_j\delta_{z_j} }[/math] be its atomic part (meaning that [math]\displaystyle{ \mu(\{z_j\})=c_j\neq 0 }[/math] and [math]\displaystyle{ \mu(\{z\})=0 }[/math] for [math]\displaystyle{ z\not\in\{z_j\} }[/math]. Then

[math]\displaystyle{ \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N|\widehat\mu(n)|^2=\sum_j|c_j|^2, }[/math]

where [math]\displaystyle{ \widehat{\mu}(n)=\int_{\mathbb T}z^{-n}\,d\mu(z) }[/math] is the [math]\displaystyle{ n }[/math]-th Fourier coefficient of [math]\displaystyle{ \mu }[/math].

• Similarly, given a real or complex Borel measure [math]\displaystyle{ \mu }[/math] on the real line [math]\displaystyle{ \mathbb R }[/math] and called [math]\displaystyle{ \mu_a=\sum_j c_j\delta_{x_j} }[/math] its atomic part, we have

[math]\displaystyle{ \lim_{R\to\infty}\frac{1}{2R}\int_{-R}^R|\widehat\mu(\xi)|^2\,d\xi=\sum_j|c_j|^2, }[/math]

where [math]\displaystyle{ \widehat{\mu}(\xi)=\int_{\mathbb R}e^{-2\pi i\xi x}\,d\mu(x) }[/math] is the Fourier transform of [math]\displaystyle{ \mu }[/math].

Proof

• First of all, we observe that if [math]\displaystyle{ \nu }[/math] is a complex measure on the circle then

[math]\displaystyle{ \frac{1}{2N+1}\sum_{n=-N}^N\widehat{\nu}(n)=\int_{\mathbb T}f_N(z)\,d\nu(z), }[/math]

with [math]\displaystyle{ f_N(z)=\frac{1}{2N+1}\sum_{n=-N}^N z^{-n} }[/math]. The function [math]\displaystyle{ f_N }[/math] is bounded by [math]\displaystyle{ 1 }[/math] in absolute value and has [math]\displaystyle{ f_N(1)=1 }[/math], while [math]\displaystyle{ f_N(z)=\frac{z^{N+1}-z^{-N}}{(2N+1)(z-1)} }[/math] for [math]\displaystyle{ z\in\mathbb{T}\setminus\{1\} }[/math], which converges to [math]\displaystyle{ 0 }[/math] as [math]\displaystyle{ N\to\infty }[/math]. Hence, by the dominated convergence theorem,

[math]\displaystyle{ \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N\widehat{\nu}(n)=\int_{\mathbb T}1_{\{1\}}(z)\,d\nu(z)=\nu(\{1\}). }[/math]

We now take [math]\displaystyle{ \mu' }[/math] to be the pushforward of [math]\displaystyle{ \overline\mu }[/math] under the inverse map on [math]\displaystyle{ \mathbb T }[/math], namely [math]\displaystyle{ \mu'(B)=\overline{\mu(B^{-1})} }[/math] for any Borel set [math]\displaystyle{ B\subseteq\mathbb T }[/math]. This complex measure has Fourier coefficients [math]\displaystyle{ \widehat{\mu'}(n)=\overline{\widehat{\mu}(n)} }[/math]. We are going to apply the above to the convolution between [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \mu' }[/math], namely we choose [math]\displaystyle{ \nu=\mu*\mu' }[/math], meaning that [math]\displaystyle{ \nu }[/math] is the pushforward of the measure [math]\displaystyle{ \mu\times\mu' }[/math] (on [math]\displaystyle{ \mathbb T\times\mathbb T }[/math]) under the product map [math]\displaystyle{ \cdot:\mathbb{T}\times\mathbb{T}\to\mathbb{T} }[/math]. By Fubini's theorem

[math]\displaystyle{ \widehat{\nu}(n)=\int_{\mathbb{T}\times\mathbb{T}}(zw)^{-n}\,d(\mu\times\mu')(z,w) =\int_{\mathbb T}\int_{\mathbb T}z^{-n}w^{-n}\,d\mu'(w)\,d\mu(z)=\widehat{\mu}(n)\widehat{\mu'}(n)=|\widehat{\mu}(n)|^2. }[/math]

So, by the identity derived earlier, [math]\displaystyle{ \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N|\widehat{\mu}(n)|^2=\nu(\{1\})=\int_{\mathbb T\times\mathbb T}1_{\{zw=1\}}\,d(\mu\times\mu')(z,w). }[/math] By Fubini's theorem again, the right-hand side equals

[math]\displaystyle{ \int_{\mathbb T}\mu'(\{z^{-1}\})\,d\mu(z)=\int_{\mathbb T}\overline{\mu(\{z\})}\,d\mu(z)=\sum_j|\mu(\{z_j\})|^2=\sum_j|c_j|^2. }[/math]

• The proof of the analogous statement for the real line is identical, except that we use the identity

[math]\displaystyle{ \frac{1}{2R}\int_{-R}^R\widehat\nu(\xi)\,d\xi=\int_{\mathbb R}f_R(x)\,d\nu(x) }[/math]

(which follows from Fubini's theorem), where [math]\displaystyle{ f_R(x)=\frac{1}{2R}\int_{-R}^R e^{-2\pi i\xi x}\,d\xi }[/math]. We observe that [math]\displaystyle{ |f_R|\le 1 }[/math], [math]\displaystyle{ f_R(0)=1 }[/math] and [math]\displaystyle{ f_R(x)=\frac{e^{2\pi iRx}-e^{-2\pi iRx}}{4\pi iRx} }[/math] for [math]\displaystyle{ x\neq 0 }[/math], which converges to [math]\displaystyle{ 0 }[/math] as [math]\displaystyle{ R\to\infty }[/math]. So, by dominated convergence, we have the analogous identity

[math]\displaystyle{ \lim_{R\to\infty}\frac{1}{2R}\int_{-R}^R\widehat\nu(\xi)\,d\xi=\nu(\{0\}). }[/math]

Consequences

• A real or complex Borel measure [math]\displaystyle{ \mu }[/math] on the circle is diffuse (i.e. [math]\displaystyle{ \mu_a=0 }[/math]) if and only if [math]\displaystyle{ \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N|\widehat\mu(n)|^2=0 }[/math].
• A probability measure [math]\displaystyle{ \mu }[/math] on the circle is a Dirac mass if and only if [math]\displaystyle{ \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N|\widehat\mu(n)|^2=1 }[/math]. (Here, the nontrivial implication follows from the fact that the weights [math]\displaystyle{ c_j }[/math] are positive and satisfy [math]\displaystyle{ 1=\sum_j c_j^2\le\sum_j c_j\le 1 }[/math], which forces [math]\displaystyle{ c_j^2=c_j }[/math] and thus [math]\displaystyle{ c_j=1 }[/math], so that there must be a single atom with mass [math]\displaystyle{ 1 }[/math].)