From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Functional equation
Date: 18 Mar 1998 05:59:28 GMT

In article <350AA2DE.D4@tirf.inpg.fr>,
Anisse TALEB  <taleb@tirf.inpg.fr> wrote:
>Well, I need help on this functional equation:
>
>	f(ax+b)=cf(x)+d    x in an interval I.
>
>and with a,b,c,d some real constants, a and c are assumed to be nonzero.
>f(x) is the unknown function.
>
>I think that this equation is related to iteration theory.

What do you think can be said about  f? Consider the case  a=1, b>0: then
your defining equation simply determines the behaviour of  f  on all of  I
from  its behaviour on  [0, b]  (or some translate thereof inside  I);
on the other hand, you've given no restrictions at all on how  f  is to
behave on that smaller interval.  Likewise if b=0 and  a>1, you may
take any  f  on  [1,a]  and then use your defining equation to extend
the definition of  f. Up to translation, these two special cases really
include all combinations of  a  and  b  except for  a=-1 (which gives
an unconstrained  f  on a half-line, extended by a kind of mirror
symmetry) and of course  a=1,b=0. (You already ruled out  a=0.)

I don't really know what "iteration theory" means -- discrete dynamical
systems, I suppose --  but yes, this seems to amount to the iterates of
linear transformations in  R^1.

Functional equations:
	index/39-XX.html
Dynamical systems:
	index/58-XX.html

dave