Convolution quotient

In mathematics, a convolution quotient is to the operation of convolution as a quotient of integers is to multiplication. Convolution quotients were introduced by Mikusiński (1949), and their theory is sometimes called Mikusiński's operational calculus.

The kind of convolution ${\textstyle (f,g)\mapsto f*g}$ with which this theory is concerned is defined by

$(f*g)(x)=\int _{0}^{x}f(u)g(x-u)\,du.$

It follows from the Titchmarsh convolution theorem that if the convolution ${\textstyle f*g}$ of two functions ${\textstyle f,g}$ that are continuous on ${\textstyle [0,+\infty )}$ is equal to 0 everywhere on that interval, then at least one of ${\textstyle f,g}$ is 0 everywhere on that interval. A consequence is that if ${\textstyle f,g,h}$ are continuous on ${\textstyle [0,+\infty )}$ then ${\textstyle h*f=h*g}$ only if ${\textstyle f=g.}$ This fact makes it possible to define convolution quotients by saying that for two functions ƒ, g, the pair (ƒ, g) has the same convolution quotient as the pair (h * ƒ,h * g).

Convolution quotients are used in an approach to making Dirac's delta function and other generalized functions logically rigorous.

References

• Mikusiński, Jan G. (1949), "Sur les fondements du calcul opératoire", Studia Math., 11: 41–70, MR 0036949
• Mikusiński, Jan (1959) [1953], Operational calculus, International Series of Monographs on Pure and Applied Mathematics, vol. 8, New York-London-Paris-Los Angeles: Pergamon Press, MR 0105594