Gunnar wrote
To begin with multiplication doesn't correspond to convolution in the discrete case but to circular convolution, so it's absolutely necessary to pad the board, possibly with reflected stones. The amount of padding needed depends on type of padding and the size of the convolution kernel. Second FFT can only be done on sizes which are composite numbers, the smaller the prime factors the better.
This second point isn't strictly true, there are fast ways of coping with smallish primes. I find references to a Winograd Fourier transform. The fact that the data is real should give a factor of 2 in time. Using a reflected 38 X 38 board with following symmetry pq bd where 'b' is the original board would mean the data is symmetrical (sort of), which might give another factor of 2. However, I share your skepticism about the usefulness of a linear influence function.
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