ABSTRACT
We begin our review of the mathematical preliminaries by discussing how we will describe some of the basic entities of Fourier analysis — numbers, functions, and operators. Perhaps the most important part of this discussion is in determining just what will be meant by the phrase “ f is a function” and by the notation “ f (x) ”. Pay close attention to this discussion even if you think you know what a “function” is. It turns out that people in different disciplines have developed slightly different views as to the meaning of this word. That is one reason a text on Fourier analysis by a mathematician specializing in, say, functional analysis will often look quite different from a corresponding text by an electrical engineer specializing in, say, signals and systems. These differences cause few problems for those who understand the differences, but they can lead the unwary into making substantially more work for themselves and even, on occasion, to making foolish errors in computations. Moreover, if we do not all agree on exactly what a function is and what f (x) denotes, then we will find it very difficult to develop clear, precise, and brief notation for the manipulations we will be doing with these things. And if we cannot adequately describe these manipulations, then the rest of this text might as well be written using grunts and hand waves.
