Pinsky phenomenon

The Pinsky phenomenon is a result in Fourier analysis, a branch of mathematics. This phenomenon was discovered by Mark Pinsky of Northwestern University. It involves the spherical inversion of the Fourier transform.

Let a function g(x) = 1 for |x| < c in 3 dimensions, with g(x) = 0 elsewhere. The jump at |x| = c will cause an oscillatory behavior of the spherical partial sums, which prevents convergence at the center of the ball as well as the possibility of Fourier inversion at x = 0. Stated differently, spherical partial sums of a Fourier integral of the indicator function of a ball are divergent at the center of the ball but convergent elsewhere to the desired indicator function. This prototype example (coined the ”Pinsky phenomenon” by Jean-Pierre Kahane, CRAS, 1995) can be suitably generalized to Fourier integral expansions in higher dimensions, both in Euclidean space and other non-compact rank-one symmetric spaces.

Notice that the lack of convergence in the Pinsky phenomenon happens far away from the boundary of the discontinuity, rather than at the discontinuity itself seen in the Gibbs phenomenon. This non-local phenomena is caused by a lensing effect.

Also related are eigenfunction expansions on a geodesic ball in a rank-one symmetric space, but one must consider boundary conditions. Pinsky and others also represent some results on the asymptotic behavior of the Fejer approximation in one dimension, inspired by work of Bump, Persi Diaconis, and J. B. Keller.