Spherical convergence of the Fourier integral of the indicator function of an $ N$-dimensional domain
Дмитрий Александрович Попов · Sbornik Mathematics · 1998
Convergence of the spherical means f{sub {omega}}(a) (here f is the characteristic function of a compact subdomain D{sup N} element of R{sup N} and {omega} is the radius of a ball in the frequency range) at a point a element of R{sup N}, a not element of {partial_derivative}D{sup N} (where {partial_derivative}D is the boundary of D{sup N}), can be characterized by the convergence exponent {sigma}(a | {partial_derivative}D{sup N}). In the case when |f{sub {omega}}(a)-f(a)|{ 0 and each {epsilon}>0 as {omega}{yields}{infinity}, {sigma}(a | {partial_derivative}D{sup N}) is the least upper bound of {gamma}. The question of the dependence of the quantity {sigma}(a | {partial_derivative}D{sup N}) on the position of the point a not element of {partial_derivative}D{sup N} and the geometry of the hypersurface {partial_derivative}D{sup N} is studied. If {partial_derivative}D{sup N} is smooth and a not element of K({partial_derivative}D{sup N}) (here K({partial_derivative}D{sup N}) is the focal surface of {partial_derivative}D{sup N}), then it is shown that {sigma}(a | {partial_derivative}D{sup N})=1 irrespective of N. A complete description of {sigma}(a | {partial_derivative}D{sup N}) for domains D{sup N} with boundary in general position and N{ =}21 there exist hypersurfaces {partial_derivative}D{sup N} in general position such that dim R({partial_derivative}D{sup N}){>=}N-21.« less