In have problems understanding connection between theory that is done in $L^2(\mathbb{R})$ and its application on discrete signal.
look at this paper http://home.ustc.edu.cn/~zhanghan/cs/Mallat_Zhang93.pdf
All theory is done for $L^2(\mathbb{R})$ and integrals, and thats all fine. On page 8(3404) functions $g$ are sampled with period $T$ so that we are able to perform calculations on a computer in order to get representation of some sampled function $f$. we use sum inner product and work with sequences instead of functions.
What allows us to use theory about square integrabile functions on sequence of numbers.
EDIT: I know that if functions have fourier transform equal to zero outside of [-W,W] (bandlimit) we can use sampling theorem to show that this $L^2$ inner product can be obtained using formula $ \langle u,v \rangle =T \sum_{l=-\infty}^{\infty} u(lT)v(lT)$ ( from book: A foundation in digital comunication by Amos Lapidoth ). If can suppose that $f$ is bandlimited but what about $g$ ? it is not bandlimited(altought it is almost zero outside [-W,W] but I ask for theoretical explanation).
I am also interested if someone can give me directions(book name, which pages) about how is same done with implementing (discrete in time) wavelet transform with Gabor wavelet or mexican hat. All book I look into everything is in $L^2$ but what is connection of that theory with implementation on discrete data.