Colombeau algebra

Colombeau algebra

In mathematics, the Colombeau algebra (named for Jean-François Colombeau) 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 (fε) of smooth functions on Rn such that

{f:} \mathbb{R}_+ \to C^\infty(\mathbb{R}^n)

(where R+=(0,∞)) is the set of "regularization" indices, and for all compact subsets K of Rn and multiindices α we have N > 0 such that

\sup_{x\in K}\left|\frac{\partial^{|\alpha|}}{(\partial x_1)^{\alpha_1}\cdots(\partial x_n)^{\alpha_n}}f_\varepsilon(x)\right| = O(\varepsilon^{-N})\qquad(\varepsilon\to 0).

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 O(εN) for all N > 0.

Embedding of distributions

The space(s) of Schwartz distributions can be embedded into this simplified algebra by (component-wise) convolution with any element of the algebra having as representative a δ-net, i.e. such that \phi_\varepsilon\to\delta in D' as ε→0.

This embedding is non-canonical, because it depends on the choice of the δ-net. However, there are versions of Colombeau algebras (so called full algebras) which allow for canonic embeddings of distributions. A well known full version is obtained by adding the mollifiers as second indexing set.

See also

References

  • Colombeau, J. F., New Generalized Functions and Multiplication of the Distributions. North Holland, Amsterdam, 1984.
  • Colombeau, J. F., Elementary introduction to new generalized functions. North-Holland, Amsterdam, 1985.
  • Nedeljkov, M., Pilipović, S., Scarpalezos, D., Linear Theory of Colombeau's Generalized Functions, Addison Wesley, Longman, 1998.
  • Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.; Geometric Theory of Generalized Functions with Applications to General Relativity, Springer Series Mathematics and Its Applications , Vol. 537, 2002; ISBN 978-1-4020-0145-1.
  • Colombeau algebra in physics


Stub icon This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.