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{{Short description|Type of mathematical distribution}}
{{distinguish|Uniform distribution (disambiguation)}}
{{inline|date=August 2019}}
In [[mathematics]], a '''homogeneous distribution''' is a [[distribution (mathematics)|distribution]] ''S'' on [[Euclidean space]] '''R'''<sup>''n''</sup> or {{nowrap|'''R'''<sup>''n''</sup> \ {0}}} that is [[homogeneous function|homogeneous]] in the sense that, roughly speaking,
:<math>S(tx) = t^m S(x)\,</math>
for all ''t'' > 0.

More precisely, let <math>\mu_t : x\mapsto x/t</math> be the scalar division operator on '''R'''<sup>''n''</sup>.  A distribution ''S'' on '''R'''<sup>''n''</sup> or {{nowrap|'''R'''<sup>''n''</sup> \ {0}}} is homogeneous of degree ''m'' provided that
:<math>S[t^{-n}\varphi\circ\mu_t] = t^mS[\varphi]</math>
for all positive real ''t'' and all test functions &phi;.  The additional factor of ''t''<sup>&minus;''n''</sup> is needed to reproduce the usual notion of homogeneity for locally integrable functions, and comes about from the [[integration by substitution|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 '''R'''<sup>''n''</sup> \ {0} to a distribution on '''R'''<sup>''n''</sup>, 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.

==Properties==
If ''S'' is a homogeneous distribution on '''R'''<sup>''n''</sup> \ {0} of degree &alpha;, then the [[weak derivative|weak]] [[partial derivative|first partial derivative]] of ''S''
:<math>\frac{\partial S}{\partial x_i}</math>
has degree &alpha;&minus;1.  Furthermore, a version of [[Euler's homogeneous function theorem]] holds: a distribution ''S'' is homogeneous of degree &alpha; if and only if
:<math>\sum_{i=1}^n x_i\frac{\partial S}{\partial x_i} = \alpha S.</math>

==One dimension==
A complete classification of homogeneous distributions in one dimension is possible.  The homogeneous distributions on {{nowrap|'''R''' \ {0}}} are given by various [[power function]]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 &minus;1.  Intuitively,
:<math>\int_{\mathbb{R}} \delta(tx)\varphi(x)\,dx = \int_{\mathbb{R}} \delta(y)\varphi(y/t)\,\frac{dy}{t} = t^{-1}\varphi(0)</math>
by making a change of variables ''y'' = ''tx'' in the "integral".  Moreover, the ''k''th weak derivative of the delta function &delta;<sup>(''k'')</sup> is homogeneous of degree &minus;''k''&minus;1.  These distributions all have support consisting only of the origin: when localized over {{nowrap|'''R''' \ {0}}}, these distributions are all identically zero.

===''x''{{su|p=&alpha;|b=+}}===
In one dimension, the function
:<math>x_+^\alpha = \begin{cases}x^\alpha&\text{if }x>0\\ 0&\text{otherwise}\end{cases}</math>
is locally integrable on {{nowrap|'''R''' \ {0}}}, and thus defines a distribution.  The distribution is homogeneous of degree &alpha;.  Similarly <math>x_-^\alpha = (-x)_+^\alpha</math> and <math>|x|^\alpha = x_+^\alpha + x_-^\alpha</math> are homogeneous distributions of degree &alpha;.

However, each of these distributions is only locally integrable on all of '''R''' provided Re(&alpha;) > &minus;1.  But although the function <math>x_+^\alpha</math> naively defined by the above formula fails to be locally integrable for Re &alpha; &le; &minus;1, the mapping
:<math>\alpha\mapsto x_+^\alpha</math>
is a [[holomorphic function]] from the right half-plane to the [[topological vector space]] of tempered distributions.  It admits a unique [[meromorphic function|meromorphic]] extension with simple poles at each negative integer {{nowrap|1=&alpha; = &minus;1, &minus;2, ...}}.  The resulting extension is homogeneous of degree &alpha;, provided &alpha; is not a negative integer, since on the one hand the relation
:<math>x_+^\alpha[\varphi\circ\mu_t] = t^{\alpha+1}x_+^\alpha[\varphi]</math>
holds and is holomorphic in &alpha; > 0.  On the other hand, both sides extend meromorphically in &alpha;, and so remain equal throughout the domain of definition.

Throughout the domain of definition, ''x''{{su|p=&alpha;|b=+}} also satisfies the following properties:
* <math>\frac{d}{dx} x_+^\alpha = \alpha x_+^{\alpha-1}</math>
* <math>x x_+^\alpha = x_+^{\alpha+1}</math>

===Other extensions===
There are several distinct ways to extend the definition of power functions to homogeneous distributions on '''R''' at the negative integers.

;&chi;{{su|p=&alpha;|b=+}}

The poles in ''x''{{su|b=+|p=&alpha;}} at the negative integers can be removed by renormalizing.  Put
:<math>\chi_+^\alpha = \frac{x_+^\alpha}{\Gamma(1+\alpha)}.</math>
This is an [[entire function]] of &alpha;.  At the negative integers,
:<math>\chi_+^{-k} = \delta^{(k-1)}.</math>
The distributions <math>\chi_+^a</math> have the properties
* <math>\frac{d}{dx} \chi_+^\alpha =  \chi_+^{\alpha-1}</math>
* <math>x \chi_+^\alpha = \alpha\chi_+^{\alpha+1}.</math>

;<math>\underline{x}^k</math>

A second approach is to define the distribution <math>\underline{x}^{-k}</math>, for {{nowrap|1=''k'' = 1, 2, ...,}}
:<math>\underline{x}^{-k} = \frac{(-1)^{k-1}}{(k-1)!}\frac{d^k}{dx^k}\log |x|.</math>
These clearly retain the original properties of power functions:
* <math>\frac{d}{dx} \underline{x}^{-k} = -k \underline{x}^{-k-1}</math>
* <math>x \underline{x}^{-k} = \underline{x}^{-k+1},\quad\text{if }k>1.</math>
These distributions are also characterized by their action on test functions
:<math>\underline{x}^{-k} = \int_{-\infty}^\infty \frac{\phi(x) - \sum_{j=0}^{k-1}x^j\phi^{(j)}(0)/j!}{x^k}\,dx,</math>
and so generalize the [[Cauchy principal value]] distribution of 1/''x'' that arises in the [[Hilbert transform]].

;(''x'' ± i0)<sup>&alpha;</sup>
Another homogeneous distribution is given by the distributional limit
:<math>(x + i0)^\alpha = \lim_{\epsilon\downarrow 0} (x+i\epsilon)^\alpha.</math>
That is, acting on test functions
:<math>(x + i0)^\alpha[\varphi] = \lim_{\epsilon\downarrow 0} \int_{\mathbb{R}} (x+i\epsilon)^\alpha\varphi(x)\,dx.</math>
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, {{nowrap|(''x'' + i0)<sup>&alpha;</sup>[&phi;]}} is an entire function of &alpha;.  Similarly,
:<math>(x-i0)^\alpha = \lim_{\epsilon\downarrow 0} (x-i\epsilon)^\alpha</math>
is also a well-defined distribution for all &alpha;

When Re &alpha; > 0,
:<math>(x\pm i0)^\alpha = x_+^\alpha + e^{\pm i\pi \alpha}x_-^\alpha,</math>
which then holds by analytic continuation whenever &alpha; is not a negative integer.  By the permanence of functional relations,
:<math>\frac{d}{dx} (x\pm i0)^\alpha =  \alpha(x\pm i0)^{\alpha-1}.</math>

At the negative integers, the identity holds (at the level of distributions on '''R''' \ {0})
:<math>(x\pm i0)^{-k}=x_+^{-k} + (-1)^kx_-^{-k}\pm\pi i(-1)^k \frac{\delta^{(k-1)}}{(k-1)!},</math>
and the singularities cancel to give a well-defined distribution on '''R'''.  The average of the two distributions agrees with <math>\underline{x}^{-k}</math>:
:<math>\frac{(x + i0)^{-k}+ (x- i0)^{-k}}{2} = \underline{x}^{-k}.</math>
The difference of the two distributions is a multiple of the delta function:
:<math>(x + i0)^{-k} - (x- i0)^{-k} = 2\pi i (-1)^k \frac{\delta^{(k-1)}}{(k-1)!},</math>
which is known as the [[Plemelj]] jump relation.

===Classification===
The following [[classification theorem]] holds {{harv|Gel'fand|Shilov|1966|loc=§3.11}}.  Let ''S'' be a distribution homogeneous of degree &alpha; on {{nowrap|'''R''' \ {0}}}.  Then <math>S = a x_+^\alpha + b x_-^\alpha</math> for some constants ''a'', ''b''.  Any distribution ''S'' on '''R''' homogeneous of degree  {{nowrap|&alpha; ≠ &minus;1, &minus;2, ...}} is of this form as well.  As a result, every homogeneous distribution of degree {{nowrap|&alpha; ≠ &minus;1, &minus;2, ...}} on {{nowrap|'''R''' \ {0}}} extends to '''R'''.

Finally, homogeneous distributions of degree &minus;''k'', a negative integer, on '''R''' are all of the form:
:<math>a\underline{x}^{-k} + b\delta^{(k-1)}.</math>

==Higher dimensions==
Homogeneous distributions on the Euclidean space {{nowrap|'''R'''<sup>''n''</sup> \ {0}}} with the origin deleted are always of the form
{{NumBlk|:|<math>S(x) = f(x/|x|)|x|^\lambda\,</math>|{{EquationRef|1}}}}
where ''&fnof;'' is a distribution on the unit sphere ''S''<sup>''n''&minus;1</sup>.  The number &lambda;, which is the degree of the homogeneous distribution ''S'', may be real or complex.

Any homogeneous distribution of the form ({{EquationNote|1}}) on {{nowrap|'''R'''<sup>''n''</sup> \ {0}}} extends uniquely to a homogeneous distribution on '''R'''<sup>''n''</sup> provided {{nowrap|Re &lambda; > &minus;''n''}}.  In fact, an analytic continuation argument similar to the one-dimensional case extends this for all {{nowrap|&lambda; ≠ &minus;''n'', &minus;''n''&minus;1, ...}}.

==References==
*{{citation|first1=I.M.|last1=Gel'fand|first2=G.E.|last2=Shilov|title=Generalized functions|volume=1|publisher=Academic Press|year=1966}}.
* {{citation|first=L.|last=Hörmander|authorlink=Lars Hörmander|title=Linear Partial Differential Operators, Volume 1|publisher=Springer-Verlag|year=1976|isbn=978-3-540-00662-6}}.
* {{citation|first=Michael|last=Taylor|title=Partial differential equations, vol. 1|publisher=Springer-Verlag|year=1996}}.

[[Category:Fourier analysis]]
[[Category:Generalized functions]]
[[Category:Schwartz distributions]]