Too early to do mathematics :-) However now you've got me going.
What I meant in my last email was nonzero absolute integral over one
period. So I think I have the careful statement of what I'm
considering.
Conjecture: Let g(x) be periodic with period p, let
integrate(abs(g(x)),x,0,p) = k let f(x) be such that
integrate(abs(f(x)),0,inf) diverges, then
integrate(abs(f(x)*g(x)),0,inf) diverges:
sketch of proof direction:
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 :-)
--
Daniel Lakeland
dlakelan at street-artists.org
http://www.street-artists.org/~dlakelan