Exercise 29, § 1

Let $f\in L^2(\mathbf{R})$ such that

a) For every integer $n\in \mathbf{N}$, the function $g_n$ defined by

$g_n(z) =(\int_{\mathbf{R}}f(x)x^n$ exp($-zx^2/2$)$dx)^2$

is holomorphic in the open set $U =\{z\in \mathbf{C}|\mathscr{R}(z)>-1\}$ of $\mathbf{C}$.

b) For every integer $n\in \mathbf{N}$, the function $h_n$ defined by

$h_n(z) =\frac{(-1)^n}{n}(\int\widehat{f}(y)y^n$ exp($-y^2/(2z)$)$dy)^2$

is holomorphic in the open set $V =\{z\in \mathbf{C}|\mathscr{R}(z^{-1})>-1\}=\{z\in \mathbf{C}|$ $|z+ 1/2|>1/2\}$ of $\mathbf{C}$.

c) There exists a holomorphic function $k_0$ in $U\cup V =\mathbf{C}-\{-1\}$ such that $k_0|U =g_0$ and $k_0|V =h_0$.

d) The function $k_0$ is zero. (Exercise prove that there exists $c\in \mathbf{C}$ such that $|k_0(z)|\leqslant$ $c/|z+ 1|$ for all $z\in \mathbf{C}-\{-1\}$ and apply Liouville's theorem to $z\mapsto (z+ 1)k_0(z)$.)

e) Now prove by induction on $n\in \mathbf{N}$ that there exists a holomorphic function $k_n$ in $U\cup V =\mathbf{C}-\{-1\}$ such that $k_n|U =g_n$ and $k_n|V =h_n$, then that $k_n= 0$. f) We have $f= 0$.

g) Let $a,b >0$ such that $ab\geqslant 1$. If $f\in L^2(\mathbf{R})$ satisfies

then $f= 0$ (“Hardy's uncertainty principle”).