ABSTRACT
Before going much further in our study of Fourier analysis, we need to develop a set of analytical tools called “identity sequences”. Except for providing approximations to the delta function (more on that in later chapters), these are not things often used in day-to-day applications. Their importance, rather, are as tools for further developing and justifying the Fourier analysis that is used in day-to-day applications. They will be essential in chapter 29 when we finally prove the few major theorems, such as the fundamental theorem on invertibility, that we could not prove earlier. In particular, using identity sequences we will derive a subtle “test for equality” that (1) will help us validate those major theorems, and (2) will provide, along with the fundamental identity from chapter 25, the starting point for the development of a much more general (and useful) theory of Fourier analysis.
