Past exam of the real and harmonic analysis course of the University of Oxford / 2008 / 3 / d

The Fourier coefficient of a measure on the circle and convolution of measures on the circle use the angular group law:Addition is modulo $2\pi$. This is the pushforward of the product complex measure under addition; equivalently its integral against a continuous $f$ is $\iint f(x+y)d\mu (x)d\nu (y)$. The integrands are bounded and the variation measures finite, so Fubini applies. In particular,Thus $\widehat{\mu \ast \nu }(n)=\hat{\mu }(n)\hat{\nu }(n)$. These measure coefficients use no additional Haar normalization factor: a probability has zeroth coefficient one.