# boundedly homogeneous function

A function  $f\!:\mathbb{R}^{n}\to\mathbb{R}$,  where $n$ is a positive integer, is called boundedly homogeneous    with respect to a set $\Lambda$ of positive reals and a real number $r$, if the equation

 $f(\lambda\vec{x})\;=\;\lambda^{r}f(\vec{x})$

is true for all and $\vec{x}\in\mathbb{R}^{n}$  and  $\lambda\in\Lambda$.  Then $\Lambda$ is the set of homogeneity and $r$ the degree of homogeneity of $f$.

Example.  The function  $x\mapsto x^{r}\sin(\ln{x})$  is boundedly homogeneous with respect to the set $\Lambda=\{e^{2\pi\nu}\,\vdots\;\;\nu\in\mathbb{Z}\}$  and  with degree of homogeneity $r$.

Let  $f\!:\mathbb{R}_{+}\to\mathbb{R}$  be a boundedly homogeneous function  with the degree of homogeneity $r$ and the set of homogeneity  $\Lambda\supset\{1\}$.  Then $f$ is of the form

 $\displaystyle f(x)\;=\;x^{r}f_{1}(\ln{x})$ (1)

where  $f_{1}\!:\mathbb{R}\to\mathbb{R}$  is a periodic real function depending on $f$.

Proof.  Defining  $g(x):=\frac{f(x)}{x^{r}}$,  we obtain

 $g(\lambda x)\;=\;\frac{f(\lambda x)}{(\lambda x)^{r}}\;=\;\frac{\lambda^{r}f(x% )}{\lambda^{r}x^{r}}\;=\;\lambda^{0}g(x)\quad\forall\lambda\in\Lambda.$

Thus $g$ is a boundedly homogeneous function with the set of homogeneity $\Lambda$ and the degree of homogeneity 0.  Moreover, define  $f_{1}(x):=g(e^{x})$.  If  $\lambda\in\Lambda\!\smallsetminus\!\{1\}$  and  $p:=\ln\lambda$,  we see that

 $f_{1}(x\!+\!p)\;=\;g(e^{x}e^{p})\;=\;g(e^{x}\lambda)\;=\;g(e^{x})\;=\;f_{1}(x)% \quad\forall x\in\mathbb{R}_{+}.$

Therefore, $f_{1}$ is periodic and (1) is in .

## References

• 1 Konrad Schlude: “Bemerkung zu beschränkt homogenen Funktionen”.  – Elemente der Mathematik 54 (1999).
Title boundedly homogeneous function BoundedlyHomogeneousFunction 2013-03-22 19:13:17 2013-03-22 19:13:17 pahio (2872) pahio (2872) 9 pahio (2872) Definition msc 26B35 msc 15-00 boundedly homogeneous set of homogeneity degree of homogeneity