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In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for treating discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering. 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.
F=F(x) to its smooth F and FG = Fsmooth Gsmooth + Fsmooth Gsingular +Fsingular Gsmooth.Such a rule applies to both the space of main functions and the space of operators which act on the space of the main functions. 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). Note that the product of singular parts does not appear in the right-hand side of (Gs(E,P)={f∈Eℕ∣∀p∈P,∃m∈ℤ:p(fn)=o(nm)}{f∈Eℕ∣∀p∈P,∀m∈ℤ:p(fn)=o(nm)}.
In particular, for (E, P)=(C,|.|) one gets (Colombeau's) generalized complex numbers (which can be "infinitely large" and "infinitesimally small" and still allow for rigorous arithmetics, very similar to nonstandard numbers). For (E, P) = (C(R),{pk}) (where pk is the supremum of all derivatives of order less than or equal to k on the ball of radius k) one gets Colombeau's simplified algebra.
This algebra "contains" all distributions T of D' via the injection
where ∗ is the convolution operation, and
This injection is non-canonical in the sense that it depends on the choice of the mollifier φ, which should be C, of integral one and have all its derivatives at 0 vanishing. 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. This means that the notion of restriction will be defined, which allows to define the support of a generalized function w.r.t. 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.
This has an especially important application in the analysis of propagation of singularities.
These include: the convolution quotient theory of Jan Mikusinski, based on the field of fractions of convolution algebras that are integral domains; and the theories of hyperfunctions, based (in their initial conception) on boundary values of analytic functions, and now making use of sheaf theory.
Bruhat introduced a class of test functions, the Schwartz–Bruhat functions, on a class of locally compact groups that goes beyond the manifolds that are the typical function domains. The applications are mostly in number theory, particularly to adelic algebraic groups. 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.
A further way in which the theory has been extended is as generalized sections of a smooth vector bundle. This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that have compact support. 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. They can be used to formulate a very general Stokes' theorem.
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