Exercise 20, § 1

Soit $n\geqslant 1$ un entier. We identify $\mathbf{R}^*_+\times \mathbf{S}_n$, and $\mathbf{R}^{n+1}-\{0\}$ by the mapping $(t, z)\mapsto tz$. Let $\omega$ be the unique measure on the sphere $\mathbf{S}_n\subset \mathbf{R}^{n+1}$ invariant under $\mathbf{O}(n+ 1,\mathbf{R})$ such that the measure $t^ndt\otimes \mu$ is identified with Lebesgue measure on $\mathbf{R}^{n+1}-\{0\}($cf. INT, VII, p. 116, exercise 8). Let $\mu$ denote the measure on $\mathbf{R}^{n+1}$ image of the measure $\omega$ by the injection of $\mathbf{S}_n$ into $\mathbf{R}^{n+1}$.

a) One has $\mu(\mathbf{S}_n) = 2\pi^{(n+1)/2}\Gamma (\frac{n+1}{2})^{-1}($cf. INT, V, §8, n$^o7)$.

b) For every real number $k\geqslant 0$, define the function $J_k:\mathbf{R}_+\rightarrow \mathbf{C}$ by

for $t\in \mathbf{R}_+$ (« Bessel function of the first species »). For every $x\in \mathbf{R}^{n+1}$, one has

where $\|x\|$ is the euclidean norm. c) If $k\in \mathbf{N}$, one has

$J_k(t) =\frac{1}{2\pi}\int_0^{2\pi}$ cos($t$ sin($u$)$-ku$)$du$

for every $t\in \mathbf{R}_+$.

d) If $n$ is even, show that $\mathscr{F}(\mu)(x)$ is a polynomial in $\|x\|^{\pm 1/2}$, cos(2$\pi \|x\|$) and sin(2$\pi \|x\|$); for $n= 2$, one has

$\mathscr{F}(\mu)(x) =$ sin(2$2\pi \pi \|x\|\|x\|$).

e) Let $g$ be a continuous function with compact support in $\mathbf{R}_+^*$. Put $f(x) =$ $g(\|x\|^2)\|x\|^{1-n/2}$ for every $x\in \mathbf{R}^n$. Then one has $\widehat{f}(y) =h(\|y\|^2)\|y\|^{1-n/2}$ for $y\in \mathbf{R}^n$, where

for every $s\in \mathbf{R}_+$ (« Hankel transform »).