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Averaging Over a Group

Suppose K is a finite cyclic group of isometries of the plane, such as a three-fold rotation group, or a group consisting of a mirror and the identity. Say K has n elements.

Suppose f is a function on the plane (real- or complex-valued). Construct a new function by averaging, for each point x in the plane, the quantities f(kx), where k varies over the group K. This means add these n quantities and divide by n; in other words, if K = {k1, . . . , kn}, then define

f*(x) =
1

n
n
S
i=1
f(kix)

The resulting function, f*, will be invariant under the group K. This is a generalization of the construction of the even part of a function on the real line: you add f(x) and f(-x) and divide by two.

If f also is invariant with respect to a translation group T, the new function will be invariant with respect to the group generated by T and K.


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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 12 Aug 1998 --- Last modified: Sep 30, 2003 9:37:05 AM
Comments to: CVM@maa.org