Spherical convergence of the~Fourier integral of the~indicator function of an~$N$-dimensional domain

D. A. Popov

Sbornik. Mathematics, Tome 189 (1998) no. 7, pp. 1101-1113

Résumé

Convergence of the spherical means $f_\Omega (a)$ (here $f$ is the characteristic function of a compact subdomain $\mathscr D^N\in \mathbb R^N$ and $\Omega$ is the radius of a ball in the frequency range) at a point $a\in \mathbb R^N$, $a\notin \partial \mathscr D^N$ (where $\partial \mathscr D$ is the boundary of $\mathscr D^N$), can be characterized by the convergence exponent $\sigma (a\,|\,\partial \mathscr D^N)$. In the case when $|f_\Omega (a)-f(a)|\leqslant O(\Omega ^{-\gamma +\varepsilon })$ for $\gamma >0$ and each $\varepsilon>0$ as $\Omega \to \infty$, $\sigma (a\,|\,\partial \mathscr D^N)$ is the least upper bound of $\gamma$. The question of the dependence of the quantity $\sigma (a\,|\,\partial \mathscr D^N)$ on the position of the point $a\notin \partial \mathscr D^N$ and the geometry of the hypersurface $\partial \mathscr D^N$ is studied. If $\partial \mathscr D^N$ is smooth and $a\notin \mathscr K(\partial \mathscr D^N)$ (here $\mathscr K(\partial \mathscr D^N)$ is the focal surface of $\partial \mathscr D^N$), then it is shown that $\sigma (a\,|\,\partial \mathscr D^N)=1$ irrespective of $N$. A complete description of $\sigma (a\,|\,\partial \mathscr D^N)$ for domains $\mathscr D^N$ with boundary in general position and $N\leqslant 10$ is given on the basis of the theory of singularities. The question of the dimension of the divergence region $\mathscr R(\partial \mathscr D^N)\in \mathscr K(\partial \mathscr D^N)$ (where the spherical means diverge as $\Omega \to \infty$) is considered. It is shown that $\dim \mathscr R(\partial \mathscr D^N)\leqslant N-3$ for $N\geqslant 3$, while for $N\geqslant 21$ there exist hypersurfaces $\partial \mathscr D^N$ in general position such that $\dim \mathscr R(\partial \mathscr D^N)\geqslant N-21$.