Can Walsh-Hadmard transform be used for convolution ?

Question

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 ?

Answer 1

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.