Exercise 2, § 1

The interval $[0,1[$ is identified with $\mathbf{T}=\mathbf{R}/\mathbf{Z}$ by the canonical projection. Let $0\leqslant a < b <1$. Let $f$ be the continuous function from $\mathbf{R}/\mathbf{Z}$ into $\mathbf{R}$ such that $f(x) = 0$ if $x \notin [a, b],f((a+b)/2) = 1$, and such that $f$ is affine on the intervals $[a,(a+b)/2]$ and $[(a+b)/2, b]$.

Calculate the Fourier transform of $f$ and prove that there exists a constant $C\geqslant 0$ such that

$|\widehat{f}(n)|\leqslant \inf\left(b-a,\frac{C}{n^2}\right)$

for all $n\in \mathbf{Z}-\{0\}$.