ABSTRACT
Recall the classical definitions of multiplication and convolution for any two piecewise continuous functions f and g on the real line: The classical product f g is the piecewise continuous function given by
f g(x) = f (x)g(x) for each x at which f and g are continuous, and the classical convolution is the function given by
f ∗ g(x) = ∫ ∞
−∞ f (x − s)g(s) 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 .
