Re: is it a quadratic function (second adjustment )

On Feb 26, 12:26 pm, tommy1729 <tommy1...@xxxxxxxxx> wrote:
robert israel wrote :



quasi <qu...@xxxxxxxx> writes:

On Tue, 26 Feb 2008 09:12:13 -0800 (PST),
Pubkeybreaker
<pubkeybrea...@xxxxxxx> wrote:

ralph0...@xxxxxxx wrote:
If f'(x)=(f(x+c)-f(x-c))/2c for some c>0 and any
x in R,

can i conclude that f is a quadratic fiction?

Yes. It is definitely fiction.

Hint: the answer is no: Consider f(x) = constant

So now revise the question ...

Prove or disprove:

If f : R --> R is a differentiable function such
that, for some
nonzero c in R, the equation
f'(x)=(f(x+c)-f(x-c))/2c holds for all x
in R, then f is a polynomial of degree at most 2.

It's not true.
By scaling, we may take c=1.
The equation r = sinh(r) has many complex solutions,
e.g. approximately
-2.7686782829874 + 7.4976762777760 i. If r is such a
solution,
f(x) = Re(exp(r x)) and f(x) = Im(exp(r x)) satisfy
your equation.
--
Robert Israel
isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics
http://www.math.ubc.ca/~israel
University of British Columbia Vancouver,
BC, Canada

trivially.

perhaps this second adjustment :

If f : R --> R is a differentiable function such that, for some nonzero c in R,
the equation f'(x)=(f(x+c)-f(x-c))/2c holds for all x in R,

AND there do not exist complex a and b such that
f(a*x + b) is a periodic function.
then f is a polynomial of degree at most 2.

Try a linear combination of solutions for different r.

Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

.