{{Multiple issues|{{improve categories|date=November 2025}} {{more sources|date=November 2025}} {{Technical|date=November 2025}} }} {{Short description|Mathematical concept}} In mathematics, a space of '''convolution quotients''' is a field of fractions of a convolution ring of functions: a convolution quotient is to the operation of convolution as a quotient of integers is to multiplication. The construction of convolution quotients allows easy algebraic representation of the Dirac delta function, integral operator, and differential operator without having to deal directly with integral transforms, which are often subject to technical difficulties with respect to whether they converge. ==Theory== Convolution quotients were introduced by {{harvs|txt|last=Mikusiński|authorlink=Jan Mikusiński|year=1949}},<ref>{{Cite journal |last=Mikusiński |first=Jan |date=1950 |title=Sur les fondements du calcul opératoire |url=http://www.impan.pl/get/doi/10.4064/sm-11-1-41-70 |journal=Studia Mathematica |language=en |volume=11 |issue=1 |pages=41–70 |doi=10.4064/sm-11-1-41-70 |issn=0039-3223}}</ref> and their theory is sometimes called ''Mikusiński's operational calculus''.
The kind of convolution <math display="inline"> (f,g)\mapsto f*g </math> with which this theory is concerned is defined by
: <math> (f*g)(x) = \int_0^x f(u) g(x-u) \, du. </math>
It follows from the Titchmarsh convolution theorem that if the convolution <math display="inline"> f*g </math> of two functions <math display="inline"> f,g</math> that are continuous on <math display="inline"> [0,+\infty) </math> is equal to 0 everywhere on that interval, then at least one of <math display="inline"> f,g</math> is 0 everywhere on that interval. A consequence is that if <math display="inline"> f,g,h</math> are continuous on <math display="inline"> [0,+\infty) </math> then <math display="inline> h*f = h*g</math> only if <math display="inline"> f = g.</math> 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 <math display="inline"> (h*f,h*g)</math>. ==Approach== As with the construction of the rational numbers from the integers, the field of convolution quotients is a direct extension of the convolution ring from which it was built. Every "ordinary" function <math>f</math> in the original space embeds canonically into the space of convolution quotients as the (equivalence class of the) pair <math>(f*g, g)</math>, in the same way that ordinary integers embed canonically into the rational numbers. Non-function elements of our new space can be thought of as "operators", or generalized functions, whose ''algebraic action on functions'' is always well-defined even if they have no representation in "ordinary" function space.
If we start with convolution ring of positive half-line functions, the above construction is identical in behavior to the Laplace transform, and ordinary Laplace-space conversion charts can be used to map expressions involving non-function operators to ordinary functions (if they exist). Yet, as mentioned above, the algebraic approach to the construction of the space bypasses the need to explicitly define the transform or its inverse, sidestepping a number of technically challenging convergence problems with the "traditional" integral transform construction.
==References== <references />
== Further reading == *{{citation|mr=0105594 |last=Mikusiński|first= Jan |title=Operational calculus |series=International Series of Monographs on Pure and Applied Mathematics|volume= 8|publisher= Pergamon Press|place= New York-London-Paris-Los Angeles|year=1959|orig-date=1953}}
Category:Generalized functions