Anisotropic dilation

For fixed real exponents $b_1,\ldots,b_d$ and $\lambda>0$ , the anisotropic dilation is

D_\lambda(y_1,\ldots,y_d)=(\lambda^{b_1}y_1,\ldots,\lambda^{b_d}y_d).

It satisfies $D_\lambda D_\mu=D_{\lambda\mu}$ , $D_\lambda^{-1}=D_{1/\lambda}$ , and has Jacobian determinant $\lambda^{b_1+\cdots+b_d}$ . The dilation is isotropic when the exponents are equal.

Homogeneous functions

A scalar function is homogeneous of degree $a$ for this dilation if $f(D_\lambda y)=\lambda^a f(y)$ . Homogeneity is relative to the chosen exponents. In particular, the same function can have different degrees for different dilations.