Homogeneous distribution

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, S ( x ) = f ( x / | x | ) | x | λ {displaystyle S(x)=f(x/|x|)|x|^{lambda },}     (1) 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. More precisely, let μ t : x ↦ x / t {displaystyle mu _{t}:xmapsto x/t} be the scalar division operator on Rn. A distribution S on Rn or Rn {0} 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. If S is a homogeneous distribution on Rn {0} of degree α, then the weak 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 A complete classification of homogeneous distributions in one dimension is possible. The homogeneous distributions on R {0} are given by various power functions. In addition to the power functions, homogeneous distributions on R include the Dirac delta function and its derivatives.