Exercise 12, § 3

Let J be the set of $f\in \mathscr{S}(\mathbf{R}^3)$ such that $\mathscr{F}(f)$ vanishes on the sphere $\mathbf{S}_2$. Let I be the set of $f\in J$ such that $(\partial /\partial y_1)(\mathscr{F}(f))$ vanishes on $\mathbf{S}_2$. Let $\overline{I}$ and $\overline{J}$ be the closures of I and J in $L^1(\mathbf{R}^3)$. Then one has $V(I) = V(J) =\mathbf{S}_2$, but $\overline{I}\not= J$. (Let $\mu$ be the positive measure of mass 1 on $\mathbf{S}_2$ invariant under the orthogonal group of $\mathbf{R}^3$; now prove using exercise 20 of II, p. 269, c) that $f(t) =t_1\mathscr{F}(\mu)(t)$ is an element of $L^{\infty}(\mathbf{R}^3)$ orthogonal to I but not to J.)