Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let F be a continuous periodic function on R^N. Let a,b be vectors in R^N. Also assume a is not parallel to b.

Does the limit of

$\varepsilon \int_0^{1/\varepsilon} F(as+b/\varepsilon) ds$

Exist as epsilon tends to 0? I think it is equal to the limit of

$\varepsilon \int_0^{1/\varepsilon} F(as) ds$

The latter limit does exist.

But I cannot prove it. ANY help would be appreciated.

share|improve this question
    
What do you mean by a periodic function, or a quasiperiodic function? What is the period? What does "quasi" mean? –  Will Sawin Jun 25 '12 at 17:27
add comment

1 Answer

up vote 1 down vote accepted

Take $a=(1,0)$, $b=(0,1)$, $F(x,y)=sin(y)$. Then $\int_0^{1/\epsilon} F(as+b/\epsilon) ds= (1/\epsilon) \sin(1/\epsilon)$, so you are asking for the limit as $\epsilon$ goes to $0$ of $\sin(1/\epsilon)$. This limit does not exist.

share|improve this answer
    
Thank you very much –  dcs24 Jun 25 '12 at 17:34
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.