I'm trying to show the orthogonality of the Haar wavelet.

$ int psi_ {jk} (x) cdot psi_ {j} (x) = 0 $

With: $ psi_ {jk} (x) = psi (2 ^ {j} x-k) = start {case} 1.0 <x

I know the case of$ j = j ^ prime, k = k ^ prime $). However, I have problems with some cases with j, k that seem to violate orthogonality.

For example: ($ j neq j ^ premium, k neq k ^ premium $), ($ j = -3, k = 0 $,$ j ^ prime = -1, k ^ prime = $ 3). This would create a situation where the waveforms are of different periods but share a common base. In that case, $ j (or $ j ^ premium $) are not completely contained in the left or right half of the other. I believe that in this case the two waveforms are no longer orthogonal?

Similarly, if I took ($ j = j ^ prime, k neq k ^ prime $), ($ j = j ^ prime = -2, k = 0, k ^ prime = 1 $). Here again, an example where one is not entirely contained in the left or right half of the other.

Evidence made online seems to use this point to prove containment and show that they are orthogonal. Why would my cases above be invalid?