Principles of Fourier Analysis, 2nd Edition

18

Integrals on Infinite Intervals

Throughout the rest of this book, a large part of our work will involve integrals over infinite intervals (usually the entire real line). While we could treat such integrals as limits of “infinite Riemann summations”, as in the previous chapter, it is much more natural (and easier) to view them as limits of integrals over finite subintervals. For example, if our interval is (−∞, ∞), then

${\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)}dx=\underset{\begin{array}{l}b\to \infty \\ a\to -\infty \end{array}}{\lim }{\displaystyle {\int }_a^{b}f\left(x\right)}dx.$

This requires, of course, that ${\displaystyle {\int }_a^{b}f\left(x\right)}dx$ exists for every finite interval (a, b) and that the above double limit exists.

Since these integrals will be so fundamental to our work, we had better discuss a few issues that could cause problems if we are not careful. The most pressing of these ...