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Haar Function


Define

 psi(x)={1   0<=x<1/2; -1   1/2<x<=1; 0   otherwise
(1)

and

 psi_(jk)(x)=psi(2^jx-k)
(2)

for j a nonnegative integer and 0<=k<=2^j-1.

HaarFns

So, for example, the first few values of psi_(jk)(x) are

psi_(00)=psi(x)
(3)
psi_(10)=psi(2x)
(4)
psi_(11)=psi(2x-1)
(5)
psi_(20)=psi(4x)
(6)
psi_(21)=psi(4x-1)
(7)
psi_(22)=psi(4x-2)
(8)
psi_(23)=psi(4x-3).
(9)

Then a function f(x) can be written as a series expansion by

 f(x)=c_0+sum_(j=0)^inftysum_(k=0)^(2^j-1)c_(jk)psi_(jk)(x).
(10)

The functions psi_(jk) and psi are all orthogonal in [0,1], with

int_0^1psi(x)psi_(jk)(x)dx=0
(11)
int_0^1psi_(jk)(x)psi_(lm)(x)dx=0
(12)

for (j,k)!=(0,0) in the first case and (j,k)!=(l,m) in the second.

These functions can be used to define wavelets. Let a function be defined on n intervals, with n a power of 2. Then an arbitrary function can be considered as an n-vector f, and the coefficients in the expansion b can be determined by solving the matrix equation

 f=W_nb
(13)

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

W_4=[1  1  1  0;  1  1 -1  0;  1 -1  0  1;  1 -1  0 -1]
(14)
=[1  1  0  0;  1 -1  0  0;  0  0  1  1;  0  0  1 -1][1  0  0  0;  0  0  1  0;  0  1  0  0;  0  0  0  1][1  1  0  0;  1 -1  0  0;  0  0  1  0;  0  0  0  1].
(15)

See also

Wavelet, Wavelet Matrix, Wavelet Transform

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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.

Referenced on Wolfram|Alpha

Haar Function

Cite this as:

Weisstein, Eric W. "Haar Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HaarFunction.html

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