Generalized function algebras
Let be an open subset of . A generalized function algebra is an associative, commutative differential algebra containing the space of distributions or other distribution spaces as a linear subspace (cf. also Generalized functions, space of). An early construction of a non-associative, non-commutative algebra was given by H. König [a6]. The main current (2000) direction has been to construct associative, commutative algebras as reduced powers of classical function spaces . A further approach uses analytic continuation and asymptotic series of distributions.
To describe the principles, consider the space of infinitely differentiable functions on (cf. also Differentiable function). Let be an infinite index set, a differential subalgebra of and a differential ideal in . The generalized function algebra is defined as the factor algebra . Assuming that is a directed set, let be a net in (cf. also Net (directed set)) converging to the Dirac measure in (cf. also Generalized functions, space of). Any compactly supported distribution can be imbedded in by convolution (cf. also Generalized function): . Appropriate conditions on and will guarantee that this extends to an imbedding of into . An imbedding of is obtained, provided the family forms a sheaf of differential algebras on (the restriction mappings are defined componentwise on representatives). This imbedding preserves the derivatives of distributions. It follows from the impossibility result of L. Schwartz (see Multiplication of distributions) that it cannot retain the pointwise product of continuous functions at the same time. If is contained in the subspace of comprised by those nets which converge weakly to zero, then an equivalence relation can be defined on by requiring that for representatives and of and . The pointwise product of continuous functions (as well as all products obtained by multiplication of distributions) are retained up to this equivalence relation. A list of typical examples of generalized function algebras follows:
1) , : there is such that for . The algebra was introduced by C. Schmieden and D. Laugwitz [a10] in their foundations of infinitesimal analysis.
2) Let be a free ultrafilter on the infinite set and define : the set of indices belongs to , let . Then is an instance of the ultrapower construction of the algebra of internal smooth functions of non-standard analysis (A. Robinson [a8]).
Neither 1) nor 2) provide sheaves on . To get a sheaf, localization must be introduced:
4) Let : for each compact subset and each multi-index there is an such that the supremum of over is of order as . Let : for each compact subset , each multi-index and each , the supremum of over is of order as . Then is one of the versions of the algebras of J.F. Colombeau [a1] (cf. also Colombeau generalized function algebras). It is distinguished by the fact that the imbedding of gives as a faithful subalgebra.
5) Let , : there is a closed, nowhere-dense subset such that for all there are a and a neighbourhood of such that for . This is the nowhere dense ideal introduced by E.E. Rosinger [a9] (cf. also Rosinger nowhere-dense generalized function algebra). The algebra contains the algebra of smooth functions defined off some nowhere-dense set as a subalgebra. Since , the imbedding of cannot be done by convolution, but uses an algebraic basis.
There are many variations on this theme, different sets , different spaces . The algebras can be defined on smooth manifolds as well. Usually, further operations can be applied to the elements of these algebras: superposition with non-linear mappings, restriction to submanifolds, pointwise evaluation (with values in the corresponding ring of constants).
The algebras offer a general framework for studying all problems involving non-linear operations, differentiation, and distributional or otherwise non-smooth data and coefficients. Applications include non-linear partial differential equations, stochastic partial differential equations, Lie symmetry transformations, distributional metrics in general relativity, quantum field theory. For a survey of current applications, see [a4].
A second approach is based on the algebras constructed by V.K. Ivanov [a5] by means of analytic or harmonic regularization of homogeneous distributions and on the weak asymptotic expansions of V.P. Maslov (see e. g. [a7]). A simple, specific example is given by the space of distributions spanned by in one dimension, where denotes the principal value distribution and the th derivative of the Dirac measure (cf. also Generalized function). Their harmonic regularizations generate a function algebra of smooth functions defined on . Each has a unique weak asymptotic expansion of the form as with coefficients in the original space ; the summation starts at some, possibly negative, . The approach was extended [a2] to the class of associated homogeneous distributions. This way the structure of an algebra may be introduced on certain subspaces of the space of asymptotic series with distribution coefficients. As an application, asymptotic solutions to non-linear partial differential equations can be constructed by direct computation with the asymptotic series.
A relation with the previous construction of generalized function algebras is obtained by observing that harmonic regularization amounts to convolution with the kernel
|[a1]||J.F. Colombeau, "New generalized functions and multiplication of distributions" , North-Holland (1984) MR0738781 Zbl 0532.46019|
|[a2]||V.G. Danilov, V.P. Maslov, V.M. Shelkovich, "Algebras of singularities of singular solutions to first-order quasilinear strictly hyperbolic systems" Theoret. Math. Phys. , 114 : 1 (1998) pp. 3–55 MR1756560|
|[a3]||Yu.V. Egorov, "A contribution to the theory of generalized functions" Russian Math. Surveys , 45 : 5 (1990) pp. 1–49 Zbl 0754.46034|
|[a4]||"Nonlinear theory of generalized functions" M. Grosser (ed.) G. Hörmann (ed.) M. Kunzinger (ed.) M. Oberguggenberger (ed.) , Chapman and Hall/CRC (1999) MR1699842 Zbl 0918.00026|
|[a5]||V.K. Ivanov, "An associative algebra of the simplest generalized functions" Sib. Math. J. , 20 (1980) pp. 509–516|
|[a6]||H. König, "Multiplikation von Distributionen I" Math. Ann. , 128 (1955) pp. 420–452 MR0068745 Zbl 0064.11303|
|[a7]||V.P. Maslov, G.A. Omel'yanov, "Asymptotic soliton-form solutions of equations with small dispersion" Russian Math. Surveys , 36 : 3 (1981) pp. 73–149 Zbl 0494.35080|
|[a8]||A. Robinson, "Non-standard analysis" , North-Holland (1966) MR0205854 Zbl 0151.00803|
|[a9]||E.E. Rosinger, "Nonlinear partial differential equations. Sequential and weak solutions" , North-Holland (1980) MR0590891 Zbl 0447.35001|
|[a10]||C. Schmieden, D. Laugwitz, "Eine Erweiterung der Infinitesimalrechnung" Math. Z. , 69 (1958) pp. 1–39 MR0095906 Zbl 0082.04203|
Generalized function algebras. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Generalized_function_algebras&oldid=24451