Sawtooth Power Series

Abstract

In this chapter, we introduce polynomials and power series expansions with respect to the triangular sine-wave. These can be used for approximations of periodic signals and unknown periodic solutions of dynamical systems. Such approximations may appear to be effective in those cases when trigonometric series converge slowly due to step-wise discontinuities or spikes. Another reason for using polynomial expansions is that they are usually more convenient for algebraic manipulations. If the process under consideration is smooth then sufficient class of smoothness of approximations is achieved by imposing specific constraints on the coefficients. Other equations for the coefficients may appear either as a result of optimization procedures, that minimize the error of approximation, or as an outcome of iterative procedures dictated by the differential equations of motion. It is also shown in this chapter that using operators Lie associated with dynamical systems essentially facilitates construction of the periodic power series.