Derivatives for multiple-valued functions induced by Galois field and Reed-Muller-Fourier expressions
Ratko Stanković, Carlos Moraga, Jaakko T. Astola · 2004
In classical mathematics, Newton-Leibniz differential operators determine the coefficients in the Taylor series. At the same time, there are relationships between the Fourier coefficients of a (differentiable) function and its derivative. By analogy, Boolean differential operators are viewed as coefficients of Taylor-MacLaurin series-like expressions for switching functions, usually denoted as Reed-Muller expressions. Spectral interpretation of these expressions, permits us to relate the Boolean difference to the coefficients in Fourier series-like expressions for switching functions. This paper considers these two possible ways of the introduction of differential operators for multiple-valued (MV) functions. We defined the logic derivatives and Gibbs derivatives for MV functions as coefficients in the Taylor-MacLaurin series for MV functions and through relationships to Fourier series-like coefficients, respectively.