VW - CVM 1.1

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, . . . , 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.

Eigenfunctions with 3-Fold Rotations