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fourier decomposition of any function f(x)?

So say we have any function f(x)

If we can write x = x0 + cos(wt), can we then deduce that f is even and periodic in t, and decompose it into an infinite sum of cosine waves like so:

f(x0 + x'cos(wt)) = sum(k=0-infinite) of [R_k(x0,x') cos(kwt)]

I am getting very confused about this notation and having trouble relating it to what I do know. Is R_k the function f in the frequency domain?

Can someone tell me if this makes sense to them, and if so, provide a brief explanation?

Thanks,

Reply 1

DFranklin

Do you mean x = x0 + cos(wt) or

x = x0 + x' cos(wt)?

If the latter, then what is x' supposed to be? (Well, in any event, what is x' supposed to be)?

Reply 2

john !!

sorry.. x' is a predetermined amplitude. So the argument of f is periodic around x0 with amplitude x'.

I am having a little trouble seeing why this makes f even in wt, (doesn't it then nessecarily need to have f(x0) = 0?), though I cam see why it may make f periodic in wt.

Sorry, I forgot the x' in the first line.

Reply 3

DFranklin

john !!

sorry.. x' is a predetermined amplitude. So the argument of f is periodic around x0 with amplitude x'.As long as x' is a constant, then that's fine.

I am having a little trouble seeing why this makes f even in wt, (doesn't it then nessecarily need to have f(x0) = 0?), though I cam see why it may make f periodic in wt.

A function f is even if f(x) = f(-x). There's no requirement for f(0) = 0. (I feel this is saying something you almost certainly know already, but I can't see any other cause for your confusion).