# Relation between discrete wavelet transform and filter banks

I have approached wavelet transformation from a projection perspective. Specifically, we can show for a certain class of functions (in the continuous setting) that they can be written in terms of a series of scaled and shifted wavelet functions:

$$f(x) = \sum_{j, k = 1}^{\infty} \langle f, \psi_{j, k} \rangle \; \psi_{j, k}(x),$$

where $$\psi$$ is a wavelet function,

$$\langle g, h \rangle := \int_{-\infty}^{\infty} g(x) \; \bar{h}(x) \; dx$$

and

$$\psi_{j, k}(x) := 2^{\frac{j}{2}} \psi \left( 2^j x - k \right).$$

Until here, it makes sense to me. However, when taking this to the discrete case, one often reads about so called $$\textit{filter banks}$$ in the literature. Specifically, it seems we iteratively convolve a signal with two different functions with subsequent down-sampling. However, in none of the literature I considered it is explained how to obtain these functions for the convolutions for a specific $$\psi$$ and I also could not find a clear explanation for why this algorithm corresponds to obtaining the coefficients $$\langle f, \psi_{j, k} \rangle$$ for different $$j, k$$. I have no prior knowledge about filter banks. Thanks.