boundedly homogeneous function

A functionMathworldPlanetmathf:n,  where n is a positive integer, is called boundedly homogeneousPlanetmathPlanetmathPlanetmathPlanetmath with respect to a set Λ of positive reals and a real number r, if the equation


is true for all and xn  and  λΛ.  Then Λ is the set of homogeneity and r the degree of homogeneity of f.

Example.  The function  xxrsin(lnx)  is boundedly homogeneous with respect to the set Λ={e2πνν}  and  with degree of homogeneity r.

TheoremMathworldPlanetmath.  Let  f:+  be a boundedly homogeneous functionMathworldPlanetmath with the degree of homogeneity r and the set of homogeneity  Λ{1}.  Then f is of the form

f(x)=xrf1(lnx) (1)

where  f1:  is a periodic real function depending on f.

Proof.  Defining  g(x):=f(x)xr,  we obtain


Thus g is a boundedly homogeneous function with the set of homogeneity Λ and the degree of homogeneity 0.  Moreover, define  f1(x):=g(ex).  If  λΛ{1}  and  p:=lnλ,  we see that


Therefore, f1 is periodic and (1) is in .


  • 1 Konrad Schlude: “Bemerkung zu beschränkt homogenen Funktionen”.  – Elemente der Mathematik 54 (1999).
Title boundedly homogeneous function
Canonical name BoundedlyHomogeneousFunction
Date of creation 2013-03-22 19:13:17
Last modified on 2013-03-22 19:13:17
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Definition
Classification msc 26B35
Classification msc 15-00
Synonym boundedly homogeneous
Defines set of homogeneity
Defines degree of homogeneity