Sum Calculus and the Discrete Transforms Methods

Abstract

In this chapter we will present some very basic elements of difference equations in Section 2.1. This is followed by a detailed discussion of the sum calculus in Section 2.2, which includes the summation by parts,and the fundamental theorem of sum calculus. These topics are essential for the sum calculus that we shall use in solving difference equations in Chapter 3, and they parallel the integration by parts and the fundamental theorem of calculus in integral calculus. Just as the integration by parts is used to derive the properties (and pairs) of the Laplace or Fourier transforms, for example, the summation by parts is essential for developing the operational properties of the discrete Fourier transforms for solving difference equations associated with boundary conditions, which is termed “the operational sum method.” This method will be introduced and illustrated in Section 2.4. Since this method parallels the Fourier integral transforms method for solving differential equations with boundary conditions, we designate Section 2.3 for a brief review of such well known integral transforms, or operational (integral) calculus method.