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]
Statementedit
Given a real or complex Borel measure on the unit circle, let be its atomic part (meaning that and for . Then
where is the -th Fourier coefficient of .
Similarly, given a real or complex Borel measure on the real line and called its atomic part, we have
First of all, we observe that if is a complex measure on the circle then
with . The function is bounded by in absolute value and has , while for , which converges to as . Hence, by the dominated convergence theorem,
We now take to be the pushforward of under the inverse map on , namely for any Borel set . This complex measure has Fourier coefficients . We are going to apply the above to the convolution between and , namely we choose , meaning that is the pushforward of the measure (on ) under the product map . By Fubini's theorem
So, by the identity derived earlier,
By Fubini's theorem again, the right-hand side equals
The proof of the analogous statement for the real line is identical, except that we use the identity
(which follows from Fubini's theorem), where .
We observe that , and for , which converges to as . So, by dominated convergence, we have the analogous identity
Consequencesedit
A real or complex Borel measure on the circle is diffuse (i.e. ) if and only if .
A probability measure on the circle is a Dirac mass if and only if . (Here, the nontrivial implication follows from the fact that the weights are positive and satisfy , which forces and thus , so that there must be a single atom with mass .)
Referencesedit
^Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle (MathOverflow)
^A complex borel measure, whose Fourier transform goes to zero (MathOverflow)