mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a space of convolution quotients is a

field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fie ...

of a convolution ring of functions: a convolution quotient is to the

operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...

convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...

as a

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s is to

multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...

. The construction of convolution quotients allows easy algebraic representation of the

Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...

integral operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms involvi ...

, and

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

without having to deal directly with

integral transforms In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily charact ...

, which are often subject to technical difficulties with respect to whether they converge. Convolution quotients were introduced by , and their theory is sometimes called ''Mikusiński's

operational calculus Operational calculus, also known as operational analysis, is a technique by which problems in Mathematical Analysis, analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomia ...

''. The kind of convolution

(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 The Titchmarsh convolution theorem describes the properties of the support of the convolution of two functions. It was proven by Edward Charles Titchmarsh in 1926. Titchmarsh convolution theorem If \varphi(t)\, and \psi(t) are integrable functio ...

that if the convolution

f*g

of two functions

f,g

that are continuous on

f,g

is 0 everywhere on that interval. A consequence is that if

f,g,h

are continuous on

[0,+\infty)

then

f = g.

This fact makes it possible to define convolution quotients by saying that for two function (mathematics)">functions ''ƒ'', ''g'', the pair (''ƒ'', ''g'') has the same convolution quotient as the pair (''h'' * ''ƒ'',''h'' * ''g''). 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

f

in the original space embeds canonically into the space of convolution quotients as the (equivalence class of the) pair

(f*g, g)

, 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 In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...

, 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

* * Generalized functions {{math-stub