Haar Function

Define

and

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

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

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

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

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

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

See also

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References

Referenced on Wolfram|Alpha

Cite this as:

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

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