Wiener's lemma explained

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 T , let \mu _{a=\sum} _{j} c _{j\delta} ^{z _{j}} be its atomic part (meaning that \mu(\{z _{j\})=c} _{j ≠} 0 and \mu(\{z\}) =0 for z\not\in\{z _{j\}} . Then\lim _{N\toinfty} (1)/(2 N +1) \sum ^{N|\widehat\mu(n)|} _{n =- N} ^{2= \sum} ^{j|c} ^{2 ,} ^{j|} where \widehat{\mu}(n) = \int _{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 R and called \mu _{a=\sum} _{j} c _{j\delta} ^{x _{j}} its atomic part, we have\lim _{R\toinfty} (1)/(2 R) \int ^{R|\widehat\mu(\xi)|} _{- R} ^{2 d\xi = \sum} ^{j|c} ^{2 ,} ^{j|} where \widehat{\mu}(\xi) = \int _{R} e ^{-2\pi} 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(1)/(2 N +1) \sum ^{N\widehat{\nu}(n) = \int} _{T} f _{N(z)d\nu(z),} with f _{N(z) = (1)/(2 N +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) = (z ^{N+1} - z ^{-N})/(( 2 N +1 )(z -1 ))} for z\in T \setminus\{ 1 \} , which converges to 0 as N\to inf ty . Hence, by the dominated convergence theorem,\lim _{N\toinfty} (1)/(2 N +1) \sum ^{N\widehat{\nu}(n) = \int} _{T} 1 _{\{1\} }(z)\,d\nu(z)=\nu(\).We now take \mu' to be the pushforward of \overline\mu under the inverse map on T , namely \mu'(B) = \overline{\mu(B ^{-1} )} for any Borel set B\subseteq 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 x \mu' (on T x T ) under the product map ⋅ : T x T \to T . By Fubini's theorem\widehat{\nu}(n) = \int _{T x T} (zw) ^{-n} d(\mu x \mu')(z,w) = \int _{T} \int _{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\toinfty} (1)/(2 N +1) \sum ^{N|\widehat{\mu}(n)|} _{n =- N} ^{2= \nu(\{ 1 \}) = \int} ^{T x T} 1 _{\{zw=1\} }\,d(\mu\times\mu')(z,w).By Fubini's theorem again, the right-hand side equals\int _{T} \mu'(\{z ^{-1} \})d\mu(z) = \int _{T} \overline{\mu(\{z\})}d\mu(z) = \sum _{j|\mu(\{z} ^{2= \sum} ^{j|c} ^{2 .} ^{j|}

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

(1)/(2 R) \int ^{R\widehat\nu(\xi)d\xi = \int} _{R} f _{R(x)d\nu(x)} (which follows from Fubini's theorem), where f _{R(x) = (1)/(2 R)} \int ^{R} _{- R} e ^{-2\pi} d\xi .We observe that |f _{R|\le} 1 , f _{R(0)=1} and f _{R(x) = (e ^{2\pi} - e ^{-2\pi})/(4 \pi iRx)} for x ≠ 0 , which converges to 0 as R\to inf ty . So, by dominated convergence, we have the analogous identity\lim _{R\toinfty} (1)/(2 R) \int ^{R\widehat\nu(\xi)d\xi = \nu(\{ 0 \}).} _{- R}

Consequences

• A real or complex Borel measure

\mu on the circle is diffuse (i.e. \mu _{a=0} ) if and only if \lim _{N\toinfty} (1)/(2 N +1) \sum ^{N|\widehat\mu(n)|} _{n =- N} ^{2=0} .\mu on the circle is a Dirac mass if and only if \lim _{N\toinfty} (1)/(2 N +1) \sum ^{N|\widehat\mu(n)|} _{n =- N} ^{2=1} . (Here, the nontrivial implication follows from the fact that the weights c _{j} are positive and satisfy 1= \sum _{j} c ^{2 \le\sum} _{j} c _{j\le} 1 , which forces c ^{2= c} _{j} and thus c _{j=1} , so that there must be a single atom with mass 1 .