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.
of all (finite) complex Borel measures on the unit circle
of continuous functions on
as its dual space.
be its discrete part (meaning that
[4][5] Similarly, on the real line
of continuous functions which vanish at infinity is the dual space of
its discrete part.
− 2 π i ξ x
is the Fourier-Stieltjes transform of
is absolutely continuous.
is absolutely continuous if its Fourier-Stieltjes sequence belongs to the sequence space
places no mass on the sets of Lebesgue measure zero (i.e.
places no mass on the countable sets.
[9] 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
, so that there must be a single atom with mass
Hence, by the dominated convergence theorem, We now take
under the inverse map on
for any Borel set
This complex measure has Fourier coefficients
We are going to apply the above to the convolution between
) 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
So, by dominated convergence, we have the analogous identity