Colombeau algebra
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.
Contents
1 Schwartz' impossibility result
2 Basic idea
3 Embedding of distributions
4 See also
5 Notes
6 References
Schwartz' impossibility result
Attempting to embed the space D′(R){displaystyle {mathcal {D}}'(mathbb {R} )}
D′(R){displaystyle {mathcal {D}}'(mathbb {R} )}such that the constant function 1{displaystyle 1}
becomes the unity in A(R){displaystyle A(mathbb {R} )}
- There is a partial derivative operator ∂{displaystyle partial }
on A(R){displaystyle A(mathbb {R} )}
- the restriction of ∂{displaystyle partial }
- the restriction of ∘{displaystyle circ }
to C(R)×C(R){displaystyle C(mathbb {R} )times C(mathbb {R} )}
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 C(R){displaystyle C(mathbb {R} )}
Colombeau algebras are constructed to satisfy conditions 1.–3. and a condition like 4., but with C(R)×C(R){displaystyle C(mathbb {R} )times C(mathbb {R} )}
Basic idea
It is defined as a quotient algebra
- CM∞(Rn)/CN∞(Rn).{displaystyle C_{M}^{infty }(mathbb {R} ^{n})/C_{N}^{infty }(mathbb {R} ^{n}).}
Here the moderate functions on Rn{displaystyle mathbb {R} ^{n}}
- CM∞(Rn){displaystyle C_{M}^{infty }(mathbb {R} ^{n})}
which are families (fε) of smooth functions on Rn{displaystyle mathbb {R} ^{n}}
- f:R+→C∞(Rn){displaystyle {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{displaystyle mathbb {R} ^{n}}
- supx∈K|∂|α|(∂x1)α1⋯(∂xn)αnfε(x)|=O(ε−N)(ε→0).{displaystyle sup _{xin 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
CN∞(Rn){displaystyle C_{N}^{infty }(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.
An introduction to Colombeau Algebras is given in here
[2]
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 ϕε→δ{displaystyle phi _{varepsilon }to delta }
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.
See also
- Generalized function
Notes
^ L. Schwartz, 1954, "Sur l'impossibilité de la multiplication des distributions", Comptes Rendus de L'Académie des Sciences 239, pp. 847–848 [1]
^ Gratus J., Colombeau Algebra: A pedagogical introduction
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; .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
ISBN 978-1-4020-0145-1. - Colombeau algebra in physics
