homogeneous function

Definition 1.

Suppose $V\mathrm{,}W$ are a vector spaces over $\mathrm{R}$, and $f\mathrm{:}V\mathrm{\to }W$ is a mapping.

• If there exists an $r\in ℝ$, such that

  $$f(\lambda v)={\lambda }^{r}f(v)$$

  for all $\lambda \in ℝ$ and $v\in V$, then $f$ is a .

• If there exists an $r\in ℝ$, such that

  $$f(\lambda v)={|\lambda |}^{r}f(v)$$

  for all $\lambda \in ℝ$ and $v\in V$, then $f$ is .

• If there exists an $r\in ℝ$, such that

  $$f(\lambda v)={\lambda }^{r}f(v)$$

  for all $\lambda \ge 0$ and $v\in V$, then $f$ is a .

Notes

For any homogeneous function as above, $f(0)=0$.

When the of homegeneity is clear one simply talks about $r$-homogeneous functions.

Title	homogeneous function
Canonical name	HomogeneousFunction
Date of creation	2013-03-22 14:44:37
Last modified on	2013-03-22 14:44:37
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	8
Author	matte (1858)
Entry type	Definition
Classification	msc 15-00
Synonym	positively homogeneous function of degree
Synonym	homogeneous function of degree
Synonym	positively homogeneous function
Related topic	HomogeneousPolynomial
Related topic	SubLinear