## Haar Function

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.

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.