Authors: Stanković, Radomir 
Moraga, Claudio
Astola, Jaakko
Title: From Fourier expansions to arithmetic-Haar expressions on quaternion groups
Journal: Applicable Algebra in Engineering, Communications and Computing
Volume: 12
Issue: 3
First page: 227
Last page: 253
Issue Date: 14-Aug-2001
Rank: M22
ISSN: 0938-1279
DOI: 10.1007/s002000100068
Arithmetic expressions for switching functions are introduced through the replacement of Boolean operations with arithmetic equivalents. In this setting, they can be regarded as the integer counterpart of Reed-Muller expressions for switching functions. However, arithmetic expressions can be interpreted as series expansions in the space of complex valued functions on finite dyadic groups in terms of a particular set of basic functions. In this case, 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 by the example of quaternion groups. Similar to the case of finite dyadic groups, 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 ensures efficiency of computation of arithmetic coefficients.
Keywords: Arithmetic expressions | Fourier transform | Haar expressions | Quaternion groups
Publisher: Springer Link

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