“Colombeau algebra”的意思、由来-开放百科全书

词条

Colombeau algebra

释义

Schwartz' impossibility result
Basic idea
Embedding of distributions
See also
Notes
References

In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this.

Such a multiplication of distributions has long been believed to be impossible because of L. Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one only wants to preserve the product of smooth functions instead such a construction becomes possible, as demonstrated first by Colombeau.

As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microlocal analysis and general relativity so far.

Schwartz' impossibility result

Attempting to embed the space of distributions on into an associative algebra , the following requirements seem to be natural:

is linearly embedded into such that the constant function becomes the unity in ,
There is a partial derivative operator on which is linear and satisfies the Leibniz rule,
the restriction of to coincides with the usual partial derivative,
the restriction of to coincides with the pointwise product.

However, L. Schwartz' result^[1] implies that these requirements cannot hold simultaneously. The same is true even if, in 4., one replaces by , the space of times continuously differentiable functions. While this result has often been interpreted as saying that a general multiplication of distributions is not possible, in fact it only states that one cannot unrestrictedly combine differentiation, multiplication of continuous functions and the presence of singular objects like the Dirac delta.

Colombeau algebras are constructed to satisfy conditions 1.–3. and a condition like 4., but with replaced by , i.e., they preserve the product of smooth (infinitely differentiable) functions only.

Basic idea

The Colombeau Algebra^[2] is defined as the quotient algebra

Here the algebra of moderate functions on is the algebra of families of smooth regularisations (f_ε)

of smooth functions on

(where R₊ = (0,∞) is the "regularization" parameter ε), such that for all compact subsets K of and all multiindices α, there is an N > 0 such that

The ideal of negligible functions is defined in the same way but with the partial derivatives instead bounded by O(ε^+N) for all N > 0.

Embedding of distributions

The space(s) of Schwartz distributions can be embedded into the simplified algebra by (component-wise) convolution with any element of the algebra having as representative a δ-net, i.e. a family of smooth functions such that 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 canonical embeddings of distributions. A well known full version is obtained by adding the mollifiers as second indexing set.

Notes

1. ^L. Schwartz, 1954, "Sur l'impossibilité de la multiplication des distributions", Comptes Rendus de L'Académie des Sciences 239, pp. 847–848
2. ^{{cite ArXiv|last = Gratus|first = J.|title = Colombeau Algebra: A pedagogical introduction| url=https://arxiv.org/abs/1308.0257|arxiv=1308.0257}}

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

3 : Smooth functions|Functional analysis|Algebras

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

Schwartz' impossibility result

Basic idea

Embedding of distributions

See also

Notes

References