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The Walsh-Hadamard transform is very fast to compute.

Can it be used to compute the convolution of two functions as it can be done with Fourier transform ?

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Not in the sense I think you mean it. First of all, the Walsh-Hadamard transform is a Fourier transform - but on the group (Z/2Z)^n instead of on the group Z/NZ. That means you can use it to compute convolutions with respect to the space of functions (Z/2Z)^n -> C. Unfortunately, unlike the case with Z/NZ you can't use this to approximate a compactly supported convolution on Z, at least not directly.

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