On the Pinsky Phenomenon

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.