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Spherical convergence of the Fourier integral of the indicator function of an $N$-dimensional domain
D. A. Popov M. V. Lomonosov Moscow State University
Abstract:
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 $\dim \mathscr R(\partial \mathscr D^N)\geqslant N-21$.
UDC:
517
MSC: Primary
42B10; Secondary
58C27 Received: 23.05.1997
DOI:
10.4213/sm342
Fulltext:
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