Define

(1) 
and

(2) 
where the Functions plotted above are
Then a Function can be written as a series expansion by

(3) 
The Functions and are all Orthogonal in ,
with

(4) 

(5) 
These functions can be used to define Wavelets. Let a Function be defined on intervals,
with a Power of 2. Then an arbitrary function can be considered as an Vector , and the
Coefficients in the expansion can be determined by solving the Matrix equation

(6) 
for , where
is the Matrix of basis functions. For example,

(7) 
The Wavelet Matrix can be computed in
steps, compared to
for the
Fourier Matrix.
See also Wavelet, Wavelet Transform
References
Haar, A. ``Zur Theorie der orthogonalen Funktionensysteme.'' Math. Ann. 69, 331371, 1910.
Strang, G. ``Wavelet Transforms Versus Fourier Transforms.'' Bull. Amer. Math. Soc. 28, 288305, 1993.
© 19969 Eric W. Weisstein
19990525