Homogeneous Function - Homogeneous Distributions

Homogeneous Distributions

A compactly supported continuous function ƒ on Rn is homogeneous of degree k if and only if

for all compactly supported test functions φ and nonzero real t. Equivalently, making a change of variable y = tx, ƒ is homogeneous of degree k if and only if

for all t and all test functions φ. The last display makes it possible to define homogeneity of distributions. A distribution S is homogeneous of degree k if

for all nonzero real t and all test functions φ. Here the angle brackets denote the pairing between distributions and test functions, and μ_t : Rn → Rn is the mapping of scalar multiplication by the real number t.