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Let be the classical system of the Walsh functions, the multiplicative semigroup of the functions represented by series of functions Wk(t)with non-negative coefficients which sum equals 1. We study the arithmetic of . The analogues of the well-known [ related to the arithmetic of the convolution semigroup of probability measures on the real line are valid in . The classes of idempotent elements, of infinitely divisible elements, of elements without indecomposable factors, and of elements without indecomposable and non-degenerate idempotent factors are completely described. We study also the class of indecomposable elements. Our method is based on the following fact: is isomorphic to the semigroup of probability measures on the groups of characters of the Cantor-Walsh group.

The paper looks at the asymptotic properties of the finite Walsh–Fourier transform applied to a discrete-time stationary time series, and shows that in many ways we have analogous results to those obtained when using the finite trigonometric Fourier transform.

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The arithmetic of a semigroup of series of Walsh functions

On the spectral decomposition of stationary time series using walsh functions. I

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2 results

The arithmetic of a semigroup of series of Walsh functions

On the spectral decomposition of stationary time series using walsh functions. I