Symmetry of Walsh Functions

by Steven H. Cullinane

For definitions of Walsh functions, see The first sixteen Walsh functions and Mathworld -- Walsh Function.

For an excellent introduction to Walsh functions, by Benjamin Jacoby (pdf format), click here.

The above illustration shows that
the first 16 Walsh functions
(from Jacoby's article)
are isomorphic to the 16 parts of the figure
in the following research note:

A related picture may be found on page 573
of A New Kind of Science, by Stephen Wolfram
(Wolfram Media, 2002):

For another picture showing the equivalence of nonconstant Walsh functions with the 15 affine hyperplanes that account for 4x4 symmetry in diamond theory, see the 15 "stencils" in Fig. VIII-8 of Shift Register Sequences, by Solomon W. Golomb, Holden-Day, 1967 (Revised edition published by Aegean Park Press, 1982).

For a bibliography of the theory of Walsh functions, click here.

For a bibliography of the applications of Walsh functions, click here.

For some philosophical remarks
related to Walsh functions, see

Time Fold.