Polycirculant matrices in discrete harmonic analysis

In this paper, we introduce a definition of a polycirculant matrix; special cases of polycirculant matrices are well-known circulant matrix and binary circulant matrix. Also, we introduce the notion of multi-convolution of discrete signals that are considered with respect to the discrete Vilenkin transform. We prove that all discrete Vilenkin functions are eigenvectors of a polycirculant matrix corresponding to eigenvalues that are discrete spectral characteristics of the original signal. This result is generalized for linear permutations of the discrete Walsh and Chrestenson transforms. Reformulating this result for multiplicative function systems, we arrive at the solution of the problem on extracting an arbitrary harmonic of the original stepped signal by an amplitude-phase operator with group phase shifts.