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:
3) Let ,
: for each compact subset
there is a
such that
for
. The algebra
was introduced by Yu.V. Egorov [a3] (cf. also Egorov generalized function algebra).
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
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References
[a1] | J.F. Colombeau, "New generalized functions and multiplication of distributions" , North-Holland (1984) |
[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 |
[a3] | Yu.V. Egorov, "A contribution to the theory of generalized functions" Russian Math. Surveys , 45 : 5 (1990) pp. 1–49 |
[a4] | "Nonlinear theory of generalized functions" M. Grosser (ed.) G. Hörmann (ed.) M. Kunzinger (ed.) M. Oberguggenberger (ed.) , Chapman and Hall/CRC (1999) |
[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 |
[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 |
[a8] | A. Robinson, "Non-standard analysis" , North-Holland (1966) |
[a9] | E.E. Rosinger, "Nonlinear partial differential equations. Sequential and weak solutions" , North-Holland (1980) |
[a10] | C. Schmieden, D. Laugwitz, "Eine Erweiterung der Infinitesimalrechnung" Math. Z. , 69 (1958) pp. 1–39 |
Generalized function algebras. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_function_algebras&oldid=16786