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 , and their theory is sometimes called "Mikusiński's operational calculus". The kind of convolution formula_1 with which this theory is concerned is defined by It follows from the Titchmarsh convolution theorem that if the convolution formula_3 of two functions formula_4 that are continuous on formula_5 is equal to 0 everywhere on that interval, then at least one of formula_4 is 0 everywhere on that interval. A consequence is that if formula_7 are continuous on  formula_5 then formula_9 only if formula_10 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.