Pinsky phenomenon

The Pinsky phenomenon is a result in Fourier analysis, a branch of mathematics.^[1] This phenomenon was discovered by Mark Pinsky of Northwestern University. It involves the spherical inversion of the Fourier transform.

Suppose n = 3 and let the function g(x) = 1 for all x such that |x| < c, with g(x) = 0 elsewhere.

This example demonstrates a phenomenon of Fourier inversion in three dimensions. The jump at |x| = c. causes an oscillatory behavior of the spherical partial sums, in particular a lack of convergence at the center of the ball:no possibility of Fourier inversion at x = 0.

Stated differently, spherical partial sums of a Fourier integral of the indicator function of a 3D ball, with ball defined in the mathematical sense, as the generalization of a circle or sphere, in three dimensions, 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), one can suitably generalize this to Fourier integral expansions in higher dimensions, both on Euclidean space and other non-compact rank-one symmetric spaces.

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.

The Pinsky phenomenon is related to, but certainly not identical to, the Gibbs phenomenon.

References

1. ↑ Taylor, Michael E. (2002). "The Gibbs phenomenon, the Pinsky phenomenon, and variants for eigenfunction expansions". Communications in Partial Differential Equations 27 (3): 565–605. doi:10.1081/PDE-120002866.