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 $\mu$ on the unit circle $\mathbb {T}$, let $\mu _{a}=\sum _{j}c_{j}\delta _{z_{j}}$ be its atomic part (meaning that $\mu (\{z_{j}\})=c_{j}\neq 0$ and $\mu (\{z\})=0$ for $z\not \in \{z_{j}\}$. Then

$\lim _{N\to \infty }{\frac {1}{2N+1}}\sum _{n=-N}^{N}|{\widehat {\mu }}(n)|^{2}=\sum _{j}|c_{j}|^{2},$

where ${\widehat {\mu }}(n)=\int _{\mathbb {T} }z^{-n}\,d\mu (z)$ is the $n$-th Fourier coefficient of $\mu$.

• Similarly, given a real or complex Borel measure $\mu$ on the real line $\mathbb {R}$ and called $\mu _{a}=\sum _{j}c_{j}\delta _{x_{j}}$ its atomic part, we have

$\lim _{R\to \infty }{\frac {1}{2R}}\int _{-R}^{R}|{\widehat {\mu }}(\xi )|^{2}\,d\xi =\sum _{j}|c_{j}|^{2},$

where ${\widehat {\mu }}(\xi )=\int _{\mathbb {R} }e^{-2\pi i\xi x}\,d\mu (x)$ is the Fourier transform of $\mu$.

Proof

• First of all, we observe that if $\nu$ is a complex measure on the circle then

${\frac {1}{2N+1}}\sum _{n=-N}^{N}{\widehat {\nu }}(n)=\int _{\mathbb {T} }f_{N}(z)\,d\nu (z),$

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

$\lim _{N\to \infty }{\frac {1}{2N+1}}\sum _{n=-N}^{N}{\widehat {\nu }}(n)=\int _{\mathbb {T} }1_{\{1\}}(z)\,d\nu (z)=\nu (\{1\}).$

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

${\widehat {\nu }}(n)=\int _{\mathbb {T} \times \mathbb {T} }(zw)^{-n}\,d(\mu \times \mu ')(z,w)=\int _{\mathbb {T} }\int _{\mathbb {T} }z^{-n}w^{-n}\,d\mu '(w)\,d\mu (z)={\widehat {\mu }}(n){\widehat {\mu '}}(n)=|{\widehat {\mu }}(n)|^{2}.$

So, by the identity derived earlier, $\lim _{N\to \infty }{\frac {1}{2N+1}}\sum _{n=-N}^{N}|{\widehat {\mu }}(n)|^{2}=\nu (\{1\})=\int _{\mathbb {T} \times \mathbb {T} }1_{\{zw=1\}}\,d(\mu \times \mu ')(z,w).$ By Fubini's theorem again, the right-hand side equals

$\int _{\mathbb {T} }\mu '(\{z^{-1}\})\,d\mu (z)=\int _{\mathbb {T} }{\overline {\mu (\{z\})}}\,d\mu (z)=\sum _{j}|\mu (\{z_{j}\})|^{2}=\sum _{j}|c_{j}|^{2}.$

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

${\frac {1}{2R}}\int _{-R}^{R}{\widehat {\nu }}(\xi )\,d\xi =\int _{\mathbb {R} }f_{R}(x)\,d\nu (x)$

(which follows from Fubini's theorem), where $f_{R}(x)={\frac {1}{2R}}\int _{-R}^{R}e^{-2\pi i\xi x}\,d\xi$. We observe that $|f_{R}|\leq 1$, $f_{R}(0)=1$ and $f_{R}(x)={\frac {e^{2\pi iRx}-e^{-2\pi iRx}}{4\pi iRx}}$ for $x\neq 0$, which converges to $0$ as $R\to \infty$. So, by dominated convergence, we have the analogous identity

$\lim _{R\to \infty }{\frac {1}{2R}}\int _{-R}^{R}{\widehat {\nu }}(\xi )\,d\xi =\nu (\{0\}).$

Consequences

• A real or complex Borel measure $\mu$ on the circle is diffuse (i.e. $\mu _{a}=0$) if and only if $\lim _{N\to \infty }{\frac {1}{2N+1}}\sum _{n=-N}^{N}|{\widehat {\mu }}(n)|^{2}=0$.
• A probability measure $\mu$ on the circle is a Dirac mass if and only if $\lim _{N\to \infty }{\frac {1}{2N+1}}\sum _{n=-N}^{N}|{\widehat {\mu }}(n)|^{2}=1$. (Here, the nontrivial implication follows from the fact that the weights $c_{j}$ are positive and satisfy $1=\sum _{j}c_{j}^{2}\leq \sum _{j}c_{j}\leq 1$, which forces $c_{j}^{2}=c_{j}$ and thus $c_{j}=1$, so that there must be a single atom with mass $1$.)