TITLE: Orthogonality of Haar wavelet functions
QUESTION [7 upvotes]: I'm reading about wavelets and I bumped into the follwing:
$\text{Haar wavelet is a step function}\; \psi(x), \text{which takes values 1 and -1, when}\; x \;\text{is in the ranges}\; [0, \frac{1}{2}) \;\text{and}\; [\frac{1}{2}, 1).$
$\text{Dilations and translations of the Haar wavelet are defined as:}$
$$\psi_{jk}(x) = \text{const} \cdot \psi(2^jx-k) $$
Now here is what I need help with :
$\text{It is apparent that:}$
$$\int\psi_{jk}(x)\cdot\psi_{j'k'}(x) = 0$$
$\text{whenever}\; j=j'\; \text{and}\; k=k'\; \text{is not satisfied simultaneously}$.
My question is: Why is the integral above true?
P.S.
here is my reference: http://gtwavelet.bme.gatech.edu/wp/kidsA.pdf (page 4)

REPLY [8 votes]: When $(j,k)\ne(j',k')$ then the two supports are either disjoint, or one of the two wavelets is constant on the support of the other.