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 discrete part. This result admits an analogous statement for measures on the real line. It was first discovered by Norbert Wiener.
Consider the space of all (finite) complex Borel measures on the unit circle and the space of continuous functions on as its dual space. Then for all and .
Given , let
be its discrete part (meaning that and for . Then
where is the -th Fourier-Stieltjes coefficient of .
Similarly, on the real line , the space of continuous functions which vanish at infinity is the dual space of and for all .
Given , let
its discrete part. Then
where is the Fourier-Stieltjes transform of .
If is continuous, then
Furthermore, tends to zero if is absolutely continuous. Equivalently, is absolutely continuous if its Fourier-Stieltjes sequence belongs to the sequence space. That is, if places no mass on the sets of Lebesgue measure zero (i.e. ), then as . Conversely, if as , then places no mass on the countable sets.
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 .
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
(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