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--- math105-s22:s:matthewk:start [2022/05/08 20:07]
matthewk [Wavelet Transform and the Uncertainty Principle]
+++ math105-s22:s:matthewk:start [2026/02/21 14:41] (current)
@@ Line 40: / Line 40: @@
 A simple example of an orthonormal wavelet is the Haar wavelet $\psi(x) = \psi_{0,0}(x) = \chi_{[0,1/2]} - \chi_{[1/2, 1]}$ where $\chi$ is the indicator function.
-We can see that wavelets allow us to localise to particular regions of our function $f$ by translating and dilating $\psi$ with $a$ and $b$. To understand how this localisation relates to the uncertainty principle, consider the orthonormal wavelet $\phi(x) = \chi_{[p,q]}(x)e^{2\pi inx}$ where we construct the family of wavelets by:
+We can see that wavelets allow us to localise to particular regions of our function $f$ by translating and dilating $\psi$ with $a$ and $b$. To understand how this localisation relates to the uncertainty principle, consider the orthonormal wavelet $\phi(x) = \chi_{[p,q]}(x)e^{2\pi inx}$ , where $p,q \in \mathbb{R}$ and $p \leq q$. We construct the family of wavelets by:
 $$\chi_{[p,q]}^{k} = \chi{[p+k(q-p), q+k(q-p)]}$$
-for $k \in \mathbb{Z}$. Note that indeed: $\langle\chi_{[p,q]}^k, \chi_{[p,q]}^l \rangle = \delta_{kl}$. By
+for $k \in \mathbb{Z}$. So we use the notation: $$\phi_{k,n} = \chi_{[p,q]}^k e^{2\pi inx}$$ Note that indeed: $\langle\phi_{k,n}, \phi_{l,m} \rangle = \delta_{kl}\delta_{nm}$.
+With this wavelet, we can analyse different sections of $f$ independently, and we can build up a picture of

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