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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.

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