Important motivations have been the technical requirements of theories of partial differential equations and group representations.
A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions.
The early history is connected with some ideas on operational calculus, and some contemporary developments are closely related to Mikio Sato's algebraic analysis.
The intensive use of the Laplace transform in engineering led to the heuristic use of symbolic methods, called operational calculus.
Since justifications were given that used divergent series, these methods were questionable from the point of view of pure mathematics.
When the Lebesgue integral was introduced, there was for the first time a notion of generalized function central to mathematics.
[1] Others proposing related theories at the time were Salomon Bochner and Kurt Friedrichs.
Its main rival in applied mathematics is mollifier theory, which uses sequences of smooth approximations (the 'James Lighthill' explanation).
[3] This theory was very successful and is still widely used, but suffers from the main drawback that distributions cannot usually be multiplied: unlike most classical function spaces, they do not form an algebra.
Another solution allowing multiplication is suggested by the path integral formulation of quantum mechanics.
Since this is required to be equivalent to the Schrödinger theory of quantum mechanics which is invariant under coordinate transformations, this property must be shared by path integrals.
[citation needed] In the first case, the multiplication is determined with some regularization of generalized function.
The associativity of multiplication is achieved; and the function signum is defined in such a way, that its square is unity everywhere (including the origin of coordinates).
Such a formalism includes the conventional theory of generalized functions (without their product) as a special case.
However, the resulting algebra is non-commutative: generalized functions signum and delta anticommute.
To obtain a canonical injection, the indexing set can be modified to be N × D(R), with a convenient filter base on D(R) (functions of vanishing moments up to order q).
If (E,P) is a (pre-)sheaf of semi normed algebras on some topological space X, then Gs(E, P) will also have this property.
a subsheaf, in particular: The Fourier transformation being (well-)defined for compactly supported generalized functions (component-wise), one can apply the same construction as for distributions, and define Lars Hörmander's wave front set also for generalized functions.
André Weil rewrote Tate's thesis in this language, characterizing the zeta distribution on the idele group; and has also applied it to the explicit formula of an L-function.
The most developed theory is that of De Rham currents, dual to differential forms.
These are homological in nature, in the way that differential forms give rise to De Rham cohomology.