Lebesgue-Riemann lamma, two-tail sums in Fourier

Jay

Re: Lebesgue-Riemann lamma, two-tail sums in Fourier

Robert Israel

From:	Jay
Subject:	Lebesgue-Riemann lamma, two-tail sums in Fourier
Date:	Mon, 24 Jan 2005 04:08:40 +0000 (UTC)

Hi: I would appreciate your help with the following:

i)Having the tail-end of a sum [0..oo] is sufficient to
have the sum converge. What are the results for sums
from -oo to +oo?. I guess we need both tails to go to
zero. Is this condition sufficient for convergence?.

ii) What if we had an alternating sum over an uncountable index?
If the sum has only non-negative terms, then it can have at
most countably many non-zero terms to be able to converge.
Is there any similar results for alternating sums over
uncountable indices?

Thanks for any help.

The Riemann-Lebesgue Lemma says that for any Leb. integrable

From:	Robert Israel
Subject:	Re: Lebesgue-Riemann lamma, two-tail sums in Fourier
Date:	24 Jan 2005 05:54:04 GMT

In article <200501240302.j0O321v09914@proapp.mathforum.org>,
Jay wrote:

> i)Having the tail-end of a sum [0..oo] is sufficient to
> have the sum converge. What are the results for sums
> from -oo to +oo?. I guess we need both tails to go to
> zero. Is this condition sufficient for convergence?.

Don't just guess, prove. It isn't hard. Hint: use epsilons
and N's.

> ii) What if we had an alternating sum over an uncountable index?
> If the sum has only non-negative terms, then it can have at
> most countably many non-zero terms to be able to converge.
> Is there any similar results for alternating sums over
> uncountable indices?

What do you mean by an alternating sum over an uncountable index?
What does convergence mean in this context?
Hint: if uncountably many entries are nonzero, then for some epsilon > 0
there are infinitely many > epsilon or infinitely many < -epsilon.

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