Exercise 64, § 1

Let $\mu$ be a measure on $\mathbf{T}$.

a) For every $x\in \mathbf{T}$, one has

1 $2i\pi hx$

$\mu(\{x\}) =$ lim $\sum\mu(h)e$.

b) Let us write $\mu=\mu_d+\mu_a$, where $\mu_a$ is an atomic measure and $\mu_d$ a diffuse measure (INT, V, §5, n$^o10$, Proposition 15). Let S be the support of $\mu_a$. One has

lim$_{H\rightarrow+\infty}\frac{1}{2H + 1}\sum|\widehat{\mu}(h)|^2=\sum|\mu(\{s\})|^2$

$|h|\leqslant Hx\in S$

(consider $(\mu*\widetilde{\mu})(\{0\})$).

c) Suppose that $\mu$ is a probability measure on $\mathbf{T}$. The following conditions are equivalent:

(i) There exists $x\in \mathbf{T}$ such that $\mu=\varepsilon_x$;

(ii) One has lim$_{|h|\rightarrow+\infty}|\widehat{\mu}(h)|^2= 1$ ;

(iii) One has

lim$_{H\rightarrow+\infty}\frac{1}{2H + 1}\sum|\widehat{\mu}(h)|^2= 1$.

$|h|\leqslant H$