Spherical convergence of the Fourier integral of the indicator function of an N-dimensional domainстатья
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Аннотация:Convergence of the spherical means f(Omega)(a) (here f is the characteristic function of a compact subdomain D-N of R-N and Omega is the radius of a ball in the frequency range) at a point a is an element of R-N, a is not an element of partial derivative D-N (where partial derivative D is the boundary of D-N), can be characterized by the convergence exponent sigma(a \ partial derivative D-N). In the case when \f(Omega)(a) - f(a)\ less than or equal to O(Omega(-gamma+epsilon)) for gamma > O and each epsilon > O as Omega --> infinity, sigma(a \ partial derivative D-N) is the least upper bound of gamma. The question of the dependence of the quantity sigma(a \ partial derivative D-N) on the position of the point a is not an element of partial derivative D-N and the geometry of the hypersurface partial derivative D-N is studied. If partial derivative D-N is smooth and a is not an element of K(partial derivative D-N) (here K(partial derivative D-N) is the focal surface of partial derivative D-N), then it is shown that sigma(a \ partial derivative D-N) = 1 irrespective of N. A complete description of sigma(a \ partial derivative D-N) for domains D-N with boundary in general position and N less than or equal to 10 is given on the basis of the theory of singularities. The question of the dimension of the divergence region R(partial derivative D-N) subset of K(partial derivative D-N) (where the spherical means diverge as Omega --> infinity) is considered. It is shown that dim R(partial derivative D-N) less than or equal to N - 3 for N greater than or equal to 3, while for N greater than or equal to 21 there exist hypersurfaces partial derivative D-N in general position such that dim R(partial derivative D-N) greater than or equal to N - 21.