Pinsky phenomenon
Encyclopedia
The Pinsky phenomenon is a result in Fourier analysis, a branch of mathematics
. This phenomenon was discovered by Mark Pinsky
of Northwestern University
in Evanston, Illinois
. 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|, 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 space
s.
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. This phenomenon was discovered by Mark Pinsky
Mark Pinsky
Mark A. Pinsky is Professor of Mathematics at Northwestern University. His research areas include probability theory, mathematical analysis, Fourier Analysis and wavelets. Pinsky earned his Ph.D at Massachusetts Institute of Technology .His published works include 125 research papersand 10 books...
of Northwestern University
Northwestern University
Northwestern University is a private research university in Evanston and Chicago, Illinois, USA. Northwestern has eleven undergraduate, graduate, and professional schools offering 124 undergraduate degrees and 145 graduate and professional degrees....
in Evanston, Illinois
Evanston, Illinois
Evanston is a suburban municipality in Cook County, Illinois 12 miles north of downtown Chicago, bordering Chicago to the south, Skokie to the west, and Wilmette to the north, with an estimated population of 74,360 as of 2003. It is one of the North Shore communities that adjoin Lake Michigan...
. It involves the spherical inversion of the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
.
Suppose n = 3 and let the function g( x) = 1 for all x such that |x|
This example demonstrates a phenomenon of Fourier
Fourier
Fourier most commonly refers to Joseph Fourier , French mathematician and physicist, or the mathematics, physics, and engineering terms named in his honor for his work on the concepts underlying them:In mathematics:...
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
Fourier
Fourier most commonly refers to Joseph Fourier , French mathematician and physicist, or the mathematics, physics, and engineering terms named in his honor for his work on the concepts underlying them:In mathematics:...
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
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
or sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, in three dimensions, are divergent at the center of the ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
but convergent elsewhere to the desired indicator function. This prototype example (coined the ”Pinsky phenomenon” by Jean-Pierre Kahane
Jean-Pierre Kahane
Jean-Pierre Kahane is a French mathematician.Kahane attended the École normale supérieure and obtained the agrégation of mathematics in 1949. He then worked for the CNRS from 1949 to 1954, first as an intern and then as a research assistant...
, CRAS, 1995), one can suitably generalize this to Fourier integral expansions in higher
dimensions, both on Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
and other non-compact rank-one symmetric space
Symmetric space
A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...
s.
Also related are eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
expansions on a geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
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
Persi Diaconis
Persi Warren Diaconis is an American mathematician and former professional magician. He is the Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University....
, and J.B.Keller.
The Pinsky phenomenon is related to, but certainly not identical to, the Gibbs phenomenon
Gibbs phenomenon
In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity: the nth partial sum of the Fourier series has large...
.