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  "body": "\n## Summary\n\nThe fast wavelet transform is the recursive filter-bank algorithm that computes the [[discrete wavelet transform|P3H4MMC4ZH]] of a signal.\nAt each level it convolves the current approximation with a low-pass and a high-pass filter and keeps every second output.\nFor a signal of length $N$ and filters of fixed length, it uses $O(N)$ operations, which is even faster than the fast Fourier transform.\nIt is also known as Mallat's algorithm, and it rests on the structure of a [[multiresolution analysis|1GHQMBR7DE]].\n\n## Explanations\n\n### L1 human\n\nStart with a list of numbers.\nReplace neighbouring pairs by a smoothed value and a difference value, which halves the length of each list.\nKeep the differences, and repeat the same step on the smoothed list.\nFor the Haar wavelet, $(4, 2, 5, 5)$ becomes averages $(6, 10)/\\sqrt{2}$ and differences $(2, 0)/\\sqrt{2}$, and then the averages are split again.\nEach level costs half as much as the one before, so the total work is about twice the cost of the first level.\n",
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