The Pascal Triangle (1654), the Reed-Muller-Fourier Transform (1992), and the Discrete Pascal Transform (2005)

This paper makes a theoretical comparative analysis of the Reed-Muller-Fourier Transform, Pascal matrices based on the Pascal triangle, and the Discrete Pascal Transform. The Reed-Muller-Fourier Transform was not originated by a Pascal matrix, however it happens to show a strong family resemblance with it, sharing several basic properties. Its area of application is the multiple-valued switching theory, mainly to obtain polynomial expressions from the value vector of multiple-valued functions. The Discrete Pascal Transform was introduced over a decade later, based on an ad hoc modification of a Pascal matrix, for applications on picture processing. It is however shown that a Discrete Pascal Transform of size p, taken modulo p equals the special Reed-Muller-Fourier Transform for the same p and n = 1. The Sierpinski fractal is close related to the Pascal matrix. Data structures based on the Sierpinski triangle have been successfully used to solve special problems in switching theory. Some of them will be addressed in the paper.