In Rudin's Functional Analysis (Chapter 7, exercise 17), Rudin claims that for $n=1$ or $2$, if $u$ is a distribution on $R^n$ with compact support $K$, whose Fourier transform $\hat{u}$ is a bounded function on $R^n$, then if $\psi \in C^\infty(R^n)$ vanishes on $K$, then $\psi u = 0$. Moreover, if $P$ is a polynomial that vanishes on $K$, then $P(-D)\hat{u} = 0$ holds. (where $D$ is the derivative operator, and $P(D)$ is just plug in $\frac{\partial }{\partial x_i}$ as $x_i$.)
But Rudin claims that if $n=3$, the above is not true. The example he gave was to consider the polynomial $1-x_1^2 -x_2^2-x_3^3$ and the distribution given by integrating against a borel probability measure, $\mu$, on $R^3$ concentrated on $S^2$ and invariant under $SO(3)$.
The fourier transform of $\mu$ is $f(x)=\sin|x|/|x|$, which does satisfy $f(x)+\Delta f(x)=0$. So I'm kind confused at what he is doing here. Any help is nice.