Fourier inversion for piecewise smooth functions in several variables

Recently we have become aware [GP, PST] of the pointwise divergence of the Fourier and Fourier-Bessel series of some piecewise constant functions in three dimensions and higher. In this note we formulate these ideas in terms of the Fourier transform. The main result gives precise necessary and sufficient conditions for the pointwise convergence of the spherical partial sums of the Fourier transform, as follows.

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