In mathematics, a homogeneous distribution is a distribution S on Euclidean space Rn or Rn \ {0} that is homogeneous in the sense that, roughly speaking, for all t > 0.
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.
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.
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.
In one dimension, the function is locally integrable on R \ {0}, and thus defines a distribution.
However, each of these distributions is only locally integrable on all of R provided Re(α) > −1.
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 α = −1, −2, ....
On the other hand, both sides extend meromorphically in α, and so remain equal throughout the domain of definition.
Throughout the domain of definition, xα+ also satisfies the following properties: There are several distinct ways to extend the definition of power functions to homogeneous distributions on R at the negative integers.
, for k = 1, 2, ..., 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.
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.
: The difference of the two distributions is a multiple of the delta function: which is known as the Plemelj jump relation.
The following classification theorem holds (Gel'fand & Shilov 1966, §3.11).
Any distribution S on R homogeneous of degree α ≠ −1, −2, ... is of this form as well.
The number λ, which is the degree of the homogeneous distribution S, may be real or complex.
In fact, an analytic continuation argument similar to the one-dimensional case extends this for all λ ≠ −n, −n−1, ....