Generalized function
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.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.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.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.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.These are homological in nature, in the way that differential forms give rise to De Rham cohomology.