Daniel Lakeland <dlakelan <at> street-artists.org> writes:
>
> On Tue, Dec 12, 2006 at 08:43:29AM -0800, Daniel Lakeland wrote:
> >
> > Too early to do mathematics However now you've got me going.
> >
> > integrate(abs(f(x) * g(x)), 0, inf) can be broken into the following
> > infinite sum
> >
> > sum(integrate(abs(g(x)*f(x)),n*p,(n+1)*p),n,0,inf)
> >
> > I think some kind of integration by parts might be helpful here.
> >
> > eventually hopefully bounding the integral by k * integrate(f(x),x,0,inf)
> >
> > of course if k = 0 then things wouldn't work.
> >
> > Don't have time to investigate further at the moment, perhaps at lunch...
> >
> > I'm glad that some of you are entertained by this
>
> I see that if it's ever to work, it's also going to require that not
> only k > 0 but also that abs(g(x) * f(x)) > 0 over some measurable
> interval within each of the sub intervals so that the two functions
> don't kill each other by careful trickery of the type being sent to me
> off list...
>
This is ultramaxima 60 seconds time:
Your theorem looks false.
The idea of a counterexample is f(x) a periodic pulse very narrow (you are
sampling g(x)) f(x) = 1 in [-e,e], 0 in the rest. Then integrate(f(x)g(x)) is
about 1/2e sum(g(x_i,1,inf) so this depends of the peeks g(x_i). So if your
sample take small values of g(x) this converges and if your sample takes the
peeks its diverge. I think you can fill the missing details.
Time to work and walk