Comment on "Computation of the Fast Walsh-Fourier Transform"1

J. L. Shanks¹has given an algorithm for computing the discrete Walsh transform (abstract Fourier transform) of a sampled periodic function whose domain of definition is the set of integers modulo 2ⁿ. An algorithm of the same efficiency, using a much simpler notation, was given for the abstract Fourier transform in my correspondence published in 1963 in this TRANSACrIONS.²This transform has an identical matrix representation; the only difference is that the function domain is represented (for computation purposes) by binary coded representations of the integers from 0 to 2ⁿ−1. These binary n-tuples form a group under vector addition, modulo two.