Principles of Fourier Analysis, 2nd Edition

4

Basic Analysis II: Integration and Infinite Series

The importance of integration and infinite series to Fourier analysis cannot be overstated. Indeed, we’ll find that the basic entities of classical Fourier analysis — Fourier transforms and Fourier series — are constructed using integrals and infinite series.

4.1    Integration

Well-Defined Integrals and Area

In this text, any reference to “an integral” will invariably be a reference to a definite integral of some function f over some interval (α, β),

${\displaystyle {\int }_{\alpha }^{\beta }f\left(x\right)\hspace{0.17em}dx}.$

(4.1)

A number of integration theories have been developed, and the precise definition of expression (4.1) and of “integrability” depends somewhat on the particular theory. For our purposes, any of the theories normally used ...