Principles of Fourier Analysis, 2nd Edition

36

Generalized Products, Convolutions and Definite Integrals

Recall the classical definitions of multiplication and convolution for any two piecewise continuous functions f and g on the real line: The classical product fg is the piecewise continuous function given by

$fg\left(x\right)=f\left(x\right)g\left(x\right)$

for each x at which f and g are continuous, and the classical convolution is the function given by

$f\ast g\left(x\right)={\displaystyle {\int }_{-\infty }^{\infty }f\left(x-s\right)g\left(s\right)ds}$

for all x in ℝ. Note that the product is always defined, while the existence of the convolution requires f(x – s)g(s) to be a “sufficiently integrable” function of s for each real value x.

Our main goal in this chapter is to describe operations for generalized functions that can be viewed as the natural generalizations of classical multiplication ...