Haar Wavelet - Haar Wavelet Properties

Haar Wavelet Properties

The Haar wavelet has several notable properties:

  1. Any continuous real function can be approximated by linear combinations of and their shifted functions. This extends to those function spaces where any function therein can be approximated by continuous functions.
  2. Any continuous real function can be approximated by linear combinations of the constant function, and their shifted functions.
  3. Orthogonality in the form
Here δi,j represents the Kronecker delta. The dual function of is itself.
4. Wavelet/scaling functions with different scale m have a functional relationship:
\begin{align} \phi(t) &= \phi(2t)+\phi(2t-1)\\ \psi(t) &= \phi(2t)-\phi(2t-1) \end{align}
5. Coefficients of scale m can be calculated by coefficients of scale m+1:
If
and
then

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