Haar Function

Define

(1)

and

(2)

for a nonnegative integer and .

So, for example, the first few values of are

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

Then a function can be written as a series expansion by

(10)

The functions and are all orthogonal in , with

			(11)
			(12)

for in the first case and in the second.

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

(13)

for , where is the matrix of basis functions. For example, the fourth-order Haar function wavelet matrix is given by

			(14)
			(15)