- 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 :\mathbb {T} \times \mathbb {T} \to \mathbb {T} }
. 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