Homogeneous distribution
Encyclopedia
In mathematics
, a homogeneous distribution is a distribution
S on Euclidean space
Rn or } that is homogeneous
in the sense that, roughly speaking,
for all t > 0.
More precisely, let
be the scalar multiplication operator on Rn. A distribution S on Rn or } is homogeneous of degree m provided that
for all positive real t and all test functions φ. The additional factor of t−n is needed to reproduce the usual notion of homogeneity for locally integrable functions, and comes about from the Jacobian change of variables
. The number m can be real or complex.
It can be a non-trivial problem to extend a given homogeneous distribution from Rn \ {0} to a distribution on Rn, although this is necessary for many of the techniques of Fourier analysis, in particular the Fourier transform
, to be brought to bear. Such an extension exists in most cases, however, although it may not be unique.
first partial derivative
of S
has degree α−1. Furthermore, a version of Euler's homogeneous function theorem holds: a distribution S is homogeneous of degree α if and only if
s. In addition to the power functions, homogeneous distributions on R include the Dirac delta function
and its derivatives.
The Dirac delta function is homogeneous of degree −1. Intuitively,
by making a change of variables y = tx in the "integral". Moreover, the kth weak derivative of the delta function δ(k) is homogeneous of degree −k−1. These distributions all have support consisting only of the origin: when localized over }, these distributions are all identically zero.
is locally integrable on }, and thus defines a distribution. The distribution is homogeneous of degree α. Similarly
and 
are homogeneous distributions of degree α.
However, each of these distributions is only locally integrable on all of R provided Re(α) > −1. But although the function
naively defined by the above formula fails to be locally integrable for Re α ≤ −1, the mapping
is a holomorphic function
from the right half-plane to the topological vector space
of tempered distributions. It admits a unique meromorphic
extension with simple poles at each negative integer . The resulting extension is homogeneous of degree α, provided α is not a negative integer, since on the one hand the relation
holds and is holomorphic in α > 0. On the other hand, both sides extend meromorphically in α, and so remain equal throughout the domain of definition.
Throughout the domain of definition, x also satisfies the following properties:
χ
The poles in x at the negative integers can be removed by renormalizing. Put
This is an entire function
of α. At the negative integers,
The distributions
have the properties
A second approach is to define the distribution
, for 
These clearly retain the original properties of power functions:
These distributions are also characterized by their action on test functions
and so generalize the Cauchy principal value
distribution of 1/x that arises in the Hilbert transform
.
(x ± i0)α
Another homogeneous distribution is given by the distributional limit
That is, acting on test functions
The branch of the logarithm is chosen to be single-valued in the upper half-plane and to agree with the natural log along the positive real axis. As the limit of entire functions, is an entire function of α. Similarly,
is also a well-defined distribution for all α
When Re α > 0,
which then holds by analytic continuation whenever α is not a negative integer. By the permanence of functional relations,
At the negative integers, the identity holds (at the level of distributions on R \ {0})
and the singularities cancel to give a well-defined distribution on R. The average of the two distributions agrees with
:
The difference of the two distributions is a multiple of the delta function:
which is known as the Plemelj jump relation.
for some constants a, b. Any distribution S on R homogeneous of degree is of this form as well. As a result, every homogeneous distribution of degree on } extends to R.
Finally, homogeneous distributions of degree −k, a negative integer, on R are all of the form:
where ƒ is a distribution on the unit sphere Sn−1. The number λ, which is the degree of the homogeneous distribution S, may be real or complex.
Any homogeneous distribution of the form on } extends uniquely to a homogeneous distribution on Rn provided . In fact, an analytic continuation argument similar to the one-dimensional case extends this for all .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a homogeneous distribution is a distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
S on Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn or } that is homogeneous
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
in the sense that, roughly speaking,
for all t > 0.
More precisely, let
be the scalar multiplication operator on Rn. A distribution S on Rn or } is homogeneous of degree m provided thatfor all positive real t and all test functions φ. The additional factor of t−n is needed to reproduce the usual notion of homogeneity for locally integrable functions, and comes about from the Jacobian change of variables
Integration by substitution
In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...
. The number m can be real or complex.
It can be a non-trivial problem to extend a given homogeneous distribution from Rn \ {0} to a distribution on Rn, although this is necessary for many of the techniques of Fourier analysis, in particular the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
, to be brought to bear. Such an extension exists in most cases, however, although it may not be unique.
Properties
If S is a homogeneous distribution on Rn \ {0} of degree α, then the weakWeak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition.- Definition :Let u be a function in the...
first partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
of S
has degree α−1. Furthermore, a version of Euler's homogeneous function theorem holds: a distribution S is homogeneous of degree α if and only if
One dimension
A complete classification of homogeneous distributions in one dimension is possible. The homogeneous distributions on } are given by various power functionPower function
In mathematics, a power function is a function of the form , where c and a are constant real numbers and x is a variable.Power functions are a special case of power law relationships, which appear throughout mathematics and statistics....
s. In addition to the power functions, homogeneous distributions on R include the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
and its derivatives.
The Dirac delta function is homogeneous of degree −1. Intuitively,
by making a change of variables y = tx in the "integral". Moreover, the kth weak derivative of the delta function δ(k) is homogeneous of degree −k−1. These distributions all have support consisting only of the origin: when localized over }, these distributions are all identically zero.
x
In one dimension, the functionis locally integrable on }, and thus defines a distribution. The distribution is homogeneous of degree α. Similarly
and are homogeneous distributions of degree α.However, each of these distributions is only locally integrable on all of R provided Re(α) > −1. But although the function
naively defined by the above formula fails to be locally integrable for Re α ≤ −1, the mappingis a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
from the right half-plane to the topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
of tempered distributions. It admits a unique meromorphic
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
extension with simple poles at each negative integer . The resulting extension is homogeneous of degree α, provided α is not a negative integer, since on the one hand the relation
holds and is holomorphic in α > 0. On the other hand, both sides extend meromorphically in α, and so remain equal throughout the domain of definition.
Throughout the domain of definition, x also satisfies the following properties:
Other extensions
There are several distinct ways to extend the definition of power functions to homogeneous distributions on R at the negative integers.χ
The poles in x at the negative integers can be removed by renormalizing. Put
This is an entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
of α. At the negative integers,
The distributions
have the properties
A second approach is to define the distribution
, for These clearly retain the original properties of power functions:
These distributions are also characterized by their action on test functions
and so generalize the Cauchy principal value
Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.-Formulation:...
distribution of 1/x that arises in the Hilbert transform
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
.
(x ± i0)α
Another homogeneous distribution is given by the distributional limit
That is, acting on test functions
The branch of the logarithm is chosen to be single-valued in the upper half-plane and to agree with the natural log along the positive real axis. As the limit of entire functions, is an entire function of α. Similarly,
is also a well-defined distribution for all α
When Re α > 0,
which then holds by analytic continuation whenever α is not a negative integer. By the permanence of functional relations,
At the negative integers, the identity holds (at the level of distributions on R \ {0})
and the singularities cancel to give a well-defined distribution on R. The average of the two distributions agrees with
:The difference of the two distributions is a multiple of the delta function:
which is known as the Plemelj jump relation.
Classification
The following classification theorem holds . Let S be a distribution homogeneous of degree α on }. Then for some constants a, b. Any distribution S on R homogeneous of degree is of this form as well. As a result, every homogeneous distribution of degree on } extends to R.Finally, homogeneous distributions of degree −k, a negative integer, on R are all of the form:
Higher dimensions
Homogeneous distributions on the Euclidean space } with the origin deleted are always of the formwhere ƒ is a distribution on the unit sphere Sn−1. The number λ, which is the degree of the homogeneous distribution S, may be real or complex.
Any homogeneous distribution of the form on } extends uniquely to a homogeneous distribution on Rn provided . In fact, an analytic continuation argument similar to the one-dimensional case extends this for all .