Preserving Essential Features in Fourier Series Approximations

[neelakash]

I hope that this is the appropriate forum to ask something about Fourier series.
My question is a little intuitive.Say I expand a function in Fourier series with
n=-∞ to n=∞.The graph of the function is available.

Now suppose,I cut off some terms for which |n|>N and expand the function.It will not be a Fourier series any more.But I am not worried about that.All I want to know whether this process is capable to preserve the essential feature of the graph.If terms like that (|n|>N) contribute very small to the actual series,then what I am telling is possible with a good approximation.Please let me know...

I encountered this problem in deriving the Parsevals formula in a Quantum Mechanics book where they have folowed the procedure in "approximation in the mean".

 

[nicktacik]

(deleted)

 

[chroot]

It's possible that the terms on the ends of the series are insigificant and can be ignored, but this is not always the case. It depends upon the function that is being transformed.

In general, though, well-behaved functions do not contain significant energy out to infinite frequencies, and very high frequency information can thus be discarded.

- Warren

 

[cliowa]

Say your function f is in C^k. Let f_n denote the n-th Fourier coefficient. Then one can prove that n^k f_n\rightarrow 0 as n\rightarrow \infty.
Is that enough for your problem?

 

[neelakash]

to nicktacik:

(deleted)

---means?

to chroot:

In general, though, well-behaved functions do not contain significant energy out to infinite frequencies, and very high frequency information can thus be discarded.

Say I am talking about a pulse like a Gaussian (momentum representation of wave function).Now take its F.T. which is also a Gaussian(wave function).Then I can assume what you say...right?

to cliowa,perhaps no.Because I am cutting an infinite limit integral into a summation with limit N.