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 = {k_{1}, .
. . , k_{n}}, then define
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.
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