Colombeau algebra

From Wikipedia, the free encyclopedia

In mathematics, the Colombeau algebra is an algebra introduced with the aim of constructing an improved theory of distributions, in which multiplication is not problematic. The origins of the theory are in applications to quasilinear hyperbolic partial differential equations.

It is defined as a quotient algebra

$C^\infty_M(\mathbb{R}^n)/C^\infty_N(\mathbb{R}^n).$

Here the moderate functions on Rⁿ are defined as

$C^\infty_M(\mathbb{R}^n)$

which are families f(x) of smooth functions on Rⁿ such that

$f:C^\infty(\mathbb{R}^n) \rightarrow \mathbb{R}^+$

(where R⁺=(0,∞)) and for all compact subsets K of Rⁿ and multiindices α we have N > 0, η > 0 and c > 0 such that for all ε > 0 with ε < η and x in K,

$\left|\frac{\partial^{|\alpha|}}{(\partial x^1)^{\alpha_1}\cdots(\partial x^n)^{\alpha_n}}f_\varepsilon(x)\right|\leq\frac{c}{\varepsilon^N}.$

The ideal $C^\infty_N(\mathbb{R}^n)$ of negligible functions is defined in the same way but with the partial derivatives instead bounded by cε^N for all N > 0.

[edit] Embedding of distributions

The space(s) of Schwartz distributions can be embedded into this simplified algebra by convolution with a net of suitable mollifiers.

(Details should be added here, from generalized functions and/or mollifiers.)

This embedding is non-canonical, because it depends on the choice of the mollifier. The "full" (non-simplified) Colombeau algebra is obtained by adding the mollifiers as second indexing set. In this setting, Schwartz distributions can be canonically embedded.