From Fourier expansions to arithmetic-Haar expressions on quaternion groups

Arithmetic expressions for switching functions are introduced through the r eplacement of Boolean operations with arithmetic equivalents. In this setti ng, they can be regarded as the integer counterpart of Reed-Muller expressi ons for switching functions. However, arithmetic expressions can be interpr eted as series expansions in the space of complex valued functions on finit e dyadic groups in terms of a particular set of basic functions. In this ca se, arithmetic expressions can be derived from the Walsh series expansions, which are the Fourier expansions on finite dyadic groups. In this paper, we extend the arithmetic expressions to non-Abelian groups b y the example of quaternion groups. Similar to the case of finite dyadic gr oups, the arithmetic expressions on quaternion groups are derived from the Fourier expansions. Attempts are done to get the related transform matrices with a structure similar to that of the Haar transform matrices, which ens ures efficiency of computation of arithmetic coefficients.