posted
I've never asked you folks for homework help before, but I guess there's a first time for everything. I figure there has got to be at least one person on this board that can help me out.
There's a proof for my applied math class that I just can't figure out. In fact, I don't even know where to start (and the problem statement even includes a hint).
The Riemann-Lebesgue Lemma is stated like so:
quote:lim (|k|->inf) { integral from a to b of [ exp(ikx) f(x) dx ] } = 0 pic
or
integral from a to b of [ exp(ikx) f(x) dx ] = o(1/k) pic
I'm to prove this for piecewise continuous functions.
The hint is: Split the interval into subintervals on which the function is continuous.
I'm totally lost and my final is tomorrow, leaving me no opportunity to further consult with my professor. Any help that could be rendered would be much appreciated.
And, yes, I'm aware that I should do my own homework.
Edit: I added some links to prettier versions of the equations.
posted
I would be happy to help if you translate the post, it appears to be written in a foreign language.
Posts: 14428 | Registered: Aug 2001
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Ok, I remember just enough calculus to understand (sort of) the question -- and have NO clue how to solve it. (Which helps about as much as the other replies to date, I'm sure.)
However, if you don't get any more helpful replies within the next hour or two, I'll see if my mom is willing to take a crack at it. Posts: 32919 | Registered: Mar 2003
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you either need to go from [a,-1][-1,1] [ 1, b] Or you might need zero in there somewhere but I don't think so it has been a long time since I did that stuff.
posted
Well, the exponential function is entire, so it should be continuous everywhere, not just piecewise. And since f(x) is a general function, I don't think the non-continuous points are specifically defined... Damn my eyes for not being better at math!
Posts: 4534 | Registered: Jan 2003
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posted
I googled as well, but that one didn't help me too much, as it proves one of the real cases of the lemma, not the complex. And I am too dumb to be able to figure out the one from the other. But thanks, just_me.
Posts: 4534 | Registered: Jan 2003
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posted
Well, my proof involves a fair amount of handwaving as I switch the order of a couple of limits and a sum without justifying that I can do that, but I think it works. Thanks folks. Now all I need is to get 117% on my final so I can get an A in the class. Posts: 4534 | Registered: Jan 2003
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posted
Good luck saxy. And rest assured, I will have plenty of stupid math questions in a few weeks when I start my last-math-class-ever.
Posts: 318 | Registered: Apr 2002
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posted
I love integrals but my thought patterns are out of practice. I keep thinking I should volunteer tutor at the local community college, but that would be a heavy time commitment.
If you still want a hint though: If f were smooth, this could be done by integration by parts. And if f isn't smooth, you can use an approximation argument to dodge that.
This should avoid any nasty sums or interchanges of limits.
Posts: 643 | Registered: Feb 2000
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