Extra Credit-discrete Fourier transform

[intelwanderer]

Homework Statement

Show that the normal coordinates for the equation we derived in Problem Set 4, Problem 2 are given by the discrete Fourier transform of an infinite series and the eigenfrequencies corresponding to each k.

http://www.ph.utexas.edu/~asimha/PHY315/Solutions-4.pdf

The solution to the problem we are solving for is given above-Problem 2

q'' = ω(2qj - qj+1 - qj-1)

Homework Equations

I'd assume we'd use the infinite Fourier transform given in the problem statement.

Ʃexp(2∏ijk)q(t) from j = negative∞ to ∞

That's all I can think of at the moment.

The Attempt at a Solution

OK, this is an extra credit problem, so naturally, it's harder than the rest of the HW. We don't have to use this kind of math on a regular basis, hence I'm more than a little lost on how to get started. I went with this over the TA weeks ago(I was curious on how you got the real solution), but was too dumb to write down what he said. I'm sure if I got a little help, I'll remember...

 

[jbsteele]

I know that the Fourier transform of an infinite series is given by the expression above, and that it can be used to solve equations such as the one we are dealing with. I think that the eigenfrequencies would come in as part of the Fourier transform, but I'm not sure how to go about applying them here. Any help would be appreciated!