Pinsky phenomenon

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The Pinsky phenomenon is a result in Fourier analysis, a branch of mathematics.

[edit] History

This mathematical phenomenon was discovered by Dr. Mark Pinsky of Northwestern University in Evanston, Illinois, near Chicago. It involves Fourier inversions. Another example of Fourier inversion is the inversion of the characteristic function of probability theory.

[edit] Description

Suppose n = 3 and let the function g( x) = 1 for all x such that −c < x < c, with g( x ) = 0 elsewhere. Compute the spherical mean, by noting that the sphere S(x, r) is contained the within the ball (mathematics) B(0, c).

This demonstrates a phenomenon of Fourier inversion in three dimensions. The jump at |x| = c. causes no possibility of Fourier inversion at x = 0.

Stated differently, spherical partial sums of a Fourier integral of the indicator function of a 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 (mathematics) 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.

[edit] Mathematical details and references

Mathematics that describe the Pinsky phenomenon are available on pages 142 to 143, and generalizations on pages 143+, in the book Introduction to Fourier Analysis and Wavelets, by Mark A. Pinsky, 2002, ISBN: 9780534376604 Publisher: Thomson Brooks/Cole.

The name Pinsky phenomenon given to this result is due to Jean-Pierre Kahane, CRAS, 1995

The result is cited widely; refer to Google Scholar.