Dirichlet multiplication and easily-generated function analysis

Abstract

Following sine-cosine functions, sawtooth wave, square wave, triangular wave, and trapezoidal wave become new easily-generated periodic functions. Can a signal be considered to be a superposition of easily-generated functions with different frequencies? In order to answer this question, we generalize Fourier analysis to easily-generated function analysis including easily-generated functions series, easily-generated function transformation, and discrete transformation for easily-generated functions. The results in this paper make it possible to represent a signal by use of easily-generated functions. A lot of techniques based on sine-cosine functions can be translated into techniques based on easily-generated functions. Because Dirichlet multiplication in number theory plays a basic role in easily-generated function analysis, we briefly introduce this concept and present a related formula. The main contents of this paper include Dirichlet multiplication and a related formula, relations between basic waveforms in electronics, easily-generated function series, easily-generated function transformation, discrete transformation for easily-generated functions, and techniques of easily-generated function analysis.