Discrete Fourier Transforms

Abstract

In Section 2.3 of Chapter 2, the notion of an integral transform was introduced in a general way. The emphasis there was on the use of such transforms, like the Laplace and Fourier transforms, in reducing differential operators to algebraic operators, hence the “operational calculus” method of solving differential equations. A special case in which the transform variable λ takes on only certain discrete values was then considered. This leads to the idea of a finite transform and the associated orthogonal expansion of a function. In Section 2.4 we took one final step, by considering the situation in which both the transform variable λ and the original independent variable are restricted to discrete sets of values. This leads to the idea of discrete transform,where, again we had our emphasis and illustrations in Section 2.4 on the use of the discrete Fourier transforms in reducing difference operators to algebraic operators, hence the “operational sum calculus” method of solving difference equations. The latter concept represents an added feature of this book.