Generalized function

"Generalized functions" are mathematical objects generalizing the notion of functions, especially in view of making also discontinuous functions indefinitely differentiable, and (which is related), to manipulate "point like charges" (densities that are zero almost everywhere but have nonzero integral).

The first realization of such a concept was the theory of distributions, developed by L. Schwartz. This theory was very successful and is still widely used, but suffers from the main drawback that it does only allow linear operations. In other words, distributions cannot be multiplied (except for very special cases): unlike most classical function spaces, they are not an algebra.

This turned into a major problem when the study of non linear problems became a more and more important issue of international research.

Several constructions of algebras of generalized functions have been proposed, among others those by E. Rosinger, Y. Egorv, R. Robinson... Today the most widely used approach to construct such associative differential algebras is based on J.-F. Colombeau's construction, see Colombeau algebra.

These are factor spaces $G=M/N$ of "moderate" modulo "negligible" nets of functions, where "moderateness" and "negligibility" refers to growth with respect to the index of the family.

A simple example is obtained by using the polynomial scale on N, $s=\{a_{m}:\mathbb {N} \to \mathbb {R} ,n\mapsto n^{m};~m\in \mathbb {Z} \}$. Then for any semi normed algebra (E,P), the factor space will be

$G_{s}(E,P)={\frac {\{f\in E^{\mathbb {N} }\mid \forall p\in P,\exists m\in \mathbb {Z} :p(f_{n})=o(n^{m})\}}{\{f\in E^{\mathbb {N} }\mid \forall p\in P,\forall m\in \mathbb {Z} :p(f_{n})=o(n^{m})\}}}.$

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 rigourous arithmethics, very similar to nonstandard numbers), and for (E,P)=(C^∞(R),{p_k}) (where p_k 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 j(T)=(φ_n*T)_n+N, where * is the convolution, and φ_n(x) = n φ(n x).

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 in 0 vanishing. To obtain a canonical injection, the indexing set can be modified to be N x D(R), with a convenient base of filters on D(R) (functions of vanishing moments up to order q).

Sheaf structure

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Microlocal analysis

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References

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