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.^[1][2]

Statement

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

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

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

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

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

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

Proof

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

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

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

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

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

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

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

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

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

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

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

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

Consequences

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

References

1. Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle (MathOverflow)
2. A complex borel measure, whose Fourier transform goes to zero (MathOverflow)