Exercise 28, § 1

a) Let $\delta >0$ be a real number. There does not exist an integrable, non-zero function $f:\mathbf{R}\rightarrow \mathbf{C}$ with compact support, such that $\widehat{f}(y) =$ O(exp($-\delta |y|$)) when $|y| \rightarrow +\infty$. (In particular, there does not exist such a function such that $\widehat{f}$ also has compact support.)

b) Let $(\varrho_n)_{n\geqslant 1}$ be a summable sequence of strictly positive real numbers. The product

$g(x) =\prod$ sin($\varrho_n\varrho \overset{n}{x}x$),

$n\geqslant 1$

defined to be equal to 1 for $x= 0$, converges absolutely and uniformly on every compact subset of $\mathbf{R}$. It defines a continuous, non-zero, even function, such that $g\in L^1(\mathbf{R})$.

c) Let $\varrho$ be the sum of the series $\sum\varrho_n$. The support of the Fourier transform of $g$ is contained in $[-\varrho , \varrho ]$.

d) Let $r >0$ be a real number. Let $\delta :\mathbf{R}_+\rightarrow \mathbf{R}_+^*$ be a decreasing function such that $\delta (y)\rightarrow 0$ when $y\rightarrow +\infty$, such that $y\mapsto \delta (y)/y$ is integrable on $[1,+\infty [$ with respect to Lebesgue measure, and such that $y\delta (y)\rightarrow +\infty$ when $y\rightarrow +\infty$. There exists a continuous function $f\in L^1(\mathbf{R})$ with compact support contained in $[-r, r]$ such that

$\widehat{f}(y) =$ O(exp($-\delta (|y|)|y|$)) when $|y| \rightarrow +\infty$. (First show that there exists a decreasing summable sequence $(\varrho_n)$ of strictly positive real numbers such that $\varrho \leqslant r$ and $\varrho_n\geqslant e\delta (n)n^{-1}$ for all sufficiently large $n$.)