I'm studying about wavelets and here is one derivation I couldn't understand:
The constant that makes this orthogonal basis orthonormal is $2^{j/2}$. Indeed, from the definition of norm$^2$ in $L^2$: $$1=(\text{const})^2\,\,\color{red}{\boxed{\color{black}{\displaystyle \int \psi^2(2^jx-k)\,dx}}}=(\text{const})^2\cdot\color{red}{\boxed{\color{black}{\displaystyle 2^{-j}\int \psi^2(t)\,dt}}}=(\text{const})^2\cdot2^{-j}.$$
So the function $\psi(x) $ is the Haar wavelet function, which takes the value $1, -1$ on the intervals $[0,\frac{1}{2})$ and $[\frac{1}{2}, 1)$ and $0$ zero everywhere else. I have highlighted the part I don't understand...
Can someone show me, why does
$$\int \psi^2(2^jx-k)\;dx = 2^{-j}\int\psi^2(t)\;dt$$
Thank you for any help. In case you need more info just let me know. Here is my reference:
http://gtwavelet.bme.gatech.edu/wp/kidsA.pdf (page 5, top)
Make substution $t=2^{j}x-k$ then $\ dt=2^{j} dx$,
$$\int \psi^2(2^jx-k)\;dx = \int \psi^2(t)\frac{dt}{2^j}= 2^{-j}\int\psi^2(t)\;dt$$