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When n > 2 it is well known that the spherical partial sums of n-fold Fourier integrals of a characteristic function of the ball D = {x : |x| 2 < 1} do not converge at the origin. In the mathematical literature this result is called "the Pinsky phenomenon". In 1993 Pinsky established necessary and sufficient conditions for a piecewise smooth function, supported on D, which guarantee the convergence at the origin its spherical partial sums. We prove this result for nonspherical partial sums, i.e. for Fourier integrals under summation over domains bounded by level surfaces of elliptic polynomials.

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We study the behavior of Fourier integrals summed by the symbols of elliptic operators and pointwise convergence of Fourier inversion. We consider generalized localization principle which in classicalLpspaces was investigated by Sjölin (1983), Carbery and Soria (1988, 1997) and Alimov (1993). Proceeding these studies, in this paper, we establish sharp conditions for generalized localization in the class of finitely supported distributions.