Exercise 4, § 1

a) Let $\alpha >0,h >0$, and let $f:\mathbf{R}\rightarrow \mathbf{R}$ be the continuous function which is zero outside the interval $]-\alpha , \alpha [$, linear in $[-\alpha ,0]$ and $[0, \alpha ]$, equal to $h$ at 0. One has :

$\mathscr{F}(f)(t) =h_2$ sin$^2(_2\pi \alpha t)$

for $t\in \mathbf{R}^*$.

b) Let $N\in \mathbf{N}$. Put $g(x) =\sum^N_{n=-N}f(x+n)$. Then :

$\mathscr{F}(g)(t) =\frac{h}{\alpha \pi^2}$ sin$^2t(_2\pi \alpha t)$ sin((2N + 1)sin($\pi t$)$\pi t$)

for $t\in \mathbf{R}^*$.

c) We choose sequences $(\alpha_i)_{i\geqslant 1},(h_i)_{\geqslant 1},(N_i)_{i\geqslant 1}$, giving, for each integer $i$, a function $g_i$ constructed by the process of b). If $\sum_ih_i<+\infty$, the series $\sum_ig_i$ converges uniformly to a continuous function G on $\mathbf{R}$.

d) If

$\sum\alpha_ih_iN_i<+\infty ,\sum\frac{h_i}{\alpha_i}$ log $N_i<+\infty ,\sum h_iN_i= +\infty$,

$iii$

then G and $\mathscr{F}(G)$ are integrable on $\mathbf{R}$, but

Now prove that there exist examples of such sequences.