distribution
Motivation
The main motivation behind distribution theory is to
extend the common linear operators on functions,
such as the derivative, convolution, and the Fourier transform
,
so that they also apply to the singular, non-smooth, or non-integrable
functions that regularly appear in both theoretical and applied
analysis
.
Distribution theory also seeks to define suitable structures
on the spaces of functions involved
to ensure the convergence of suitable approximating functions,
and the continuity of certain operators
.
For example, the limit of derivatives should be equal
to the derivative of the limit, with some definition of the limiting
operation
.
When this program is carried out,
inevitably we find that we have to enlarge the space of objects that we
would consider as “functions”. For example, the derivative of a step
function is the Dirac delta function with a spike at the discontinuous
step;
the Fourier transform of a constant function is also a Dirac delta
function, with the spike representing infinite
spectral magnitude
at one single frequency. (These facts, of course, had long been
used in engineering mathematics.)
Remark:
Dirac’s notion of delta distributions was introduced to facilitate computations in Quantum Mechanics,
however without having at the beginning a proper mathematical definition. In part as
a (negative) reaction to such a state of affairs, von Neumann produced a mathematically
well-defined foundation of Quantum Mechanics (http://planetmath.org/QuantumGroupsAndVonNeumannAlgebras) based on actions of
self-adjoint operators on Hilbert spaces which is still currently in use, with several significant
additions such as Frechét nuclear spaces and quantum groups
.
There are several theories of such ‘generalized functions’. In this entry, we describe Schwartz’ theory of distributions, which is probably the most widely used.
Essentially, a distribution on is a linear mapping that takes a
smooth function (with compact support) on into a real number.
For example, the delta distribution is the map,
while any smooth function on induces a distribution
Distributions are also well behaved under coordinate changes, and
can be defined onto a manifold. Differential forms with
distribution valued coefficients are called currents.
However, it is not possible to define a product of two
distributions generalizing the product of usual functions.
Formal definition
A note on notation. In distribution theory, the
topological vector space of smooth functions with compact support on
an open set
is traditionally denoted by . Let us also denote by
the subset of of functions with support
in a
compact set .
Definition 1 (Distribution).
A distribution is a linear continuous functional on , i.e., a linear continuous mapping . The set of all distributions on is denoted by .
Suppose is a linear functional on .
Then is continuous
if and only if is continuous
in the origin (see this page (http://planetmath.org/ContinuousLinearMapping)).
This condition can be rewritten in various ways, and
the below theorem gives two convenient conditions that can be used to prove
that a linear mapping is a distribution.
Theorem 1.
Let be an open set in ,
and let be a linear functional on . Then the
following are equivalent:
-
1.
is a distribution.
-
2.
If is a compact set in , and be a sequence in , such that for any multi-index , we have
in the supremum norm
as , then in .
-
3.
For any compact set in , there are constants and such that for all , we have
(1) where is a multi-index, and is the supremum norm.
Proof The equivalence of (2) and (3) can be found on this page (http://planetmath.org/EquivalenceOfConditions2And3), and the equivalence of (1) and (3) is shown in [1].
Distributions of order
If is a distribution on an open set ,
and the same can be used for any
in the above inequality, then is a
distribution of order .
The set of all such distributions is denoted by .
Both usual functions and the delta distribution are of order . One can also show that by differentiating a distribution its order increases by at most one. Thus, in some sense, the order is a measure of how ”smooth” a distribution is.
Topology for
The standard topology for is the weak topology.
In this topology, a sequence of distributions
(in ) converges
to a distribution if and only if
for every .
Notes
A common notation for the action of a distribution onto a test function (i.e., with above notation) is . The motivation for this comes from this example (http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution).
References
-
1
W. Rudin, Functional Analysis
, McGraw-Hill Book Company, 1973.
- 2 L. Hrmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
Title | distribution |
Canonical name | Distribution |
Date of creation | 2013-03-22 13:44:08 |
Last modified on | 2013-03-22 13:44:08 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 23 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 46-00 |
Classification | msc 46F05 |
Synonym | ‘generalized function’ |
Related topic | ExampleOfDiracSequence |
Related topic | DiracDeltaFunction |
Related topic | DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom |
Related topic | QuantumGroups |
Related topic | FourierStieltjesAlgebraOfAGroupoid |
Related topic | QuantumOperatorAlgebrasInQuantumFieldTheories |
Related topic | QFTOrQuantumFieldTheories |
Related topic | QuantumGroup |
Defines | distribution of finite order |