Hatrack River Forum: Math Question (or, saxon75's Unbearable Shame)

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Posted by saxon75 (Member # 4589) on :

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.

[ May 18, 2004, 11:55 PM: Message edited by: saxon75 ]

Posted by Belle (Member # 2314) on :

I would be happy to help if you translate the post, it appears to be written in a foreign language.

Posted by Raia (Member # 4700) on :

Ditto...

I got a headache just READING your post! Imagine trying to understand it! [Eek!]

Posted by Jon Boy (Member # 4284) on :

Sometimes I really miss calculus.

And sometimes I don't.

Posted by Space Opera (Member # 6504) on :

I think the answer is 2. Of course, I also think that I'm an english major precisely because I suck at math.

space opera

Posted by rivka (Member # 4859) on :

My eyes! My eyes!

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. [Smile]

Posted by BannaOj (Member # 3206) on :

I think for the peicewise part

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.

AJ

Posted by just_me (Member # 3302) on :

Here's a PDF from Rutgers with a proof (I think, I haven't really examined it)

http://www.math.rutgers.edu/courses/423/423-f01/S1/riem-leb.pdf

(I just googled Riemann-Lebesgue Lemma proof )

hope this helps

good luck tomorrow!

Posted by saxon75 (Member # 4589) on :

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!

Posted by saxon75 (Member # 4589) on :

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.

Posted by saxon75 (Member # 4589) on :

Turns out I'm not quit dumb enough not to connect the real case to the complex case (which turns out to be easier).

Right on!

Posted by rivka (Member # 4859) on :

So you solved it? Excellent! *thumbsup*

Posted by Dagonee (Member # 5818) on :

Gratz! But this is why I went back for law, not science.

Operations Research, Linear Program, Algorithmic Analysis, Numerical Analysis: these I can do.

Throw an integral at me and I run screaming for the hills.

Dagonee
P.S., Good luck on your test!

Posted by saxon75 (Member # 4589) on :

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. [Smile]

Posted by Farmgirl (Member # 5567) on :

Where was Hobbes in this thread? Sounds like this question would have been right up his alley...

Glad you figured it out, Saxon. Good luck on your final.

FG

Posted by peter the bookie (Member # 3270) on :

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.

Posted by BannaOj (Member # 3206) on :

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.

AJ

Posted by BannaOj (Member # 3206) on :

btw I think of this book every time I see your screen name, saxy! (yes I know it is 76 instead of 75, but it is pretty darn close!)

AJ

[ May 19, 2004, 12:43 PM: Message edited by: BannaOj ]

Posted by saxon75 (Member # 4589) on :

[Smile]

Posted by rivka (Member # 4859) on :

LOL! I have that book somewhere or other, I think. [Big Grin]

Posted by dspeyer (Member # 758) on :

It sounds like all is solved already.

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.

Posted by dspeyer (Member # 758) on :

PS: I'm almost sure that should be O(1/k), not o(1/k)

[ May 19, 2004, 11:17 PM: Message edited by: dspeyer ]