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, 331-371, 1910.
Strang, G. ``Wavelet Transforms Versus Fourier Transforms.'' Bull. Amer. Math. Soc. 28, 288-305, 1993.
© 1996-9 Eric W. Weisstein
1999-05-25