Fourier coefficients

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Compute the sine coefficients for [itex]f(x)=e^{-x^{2}}[/itex] on the interval [itex][0,2\pi][/itex]. Does this mean [itex]f(x+2\pi k)=f(x)[/itex], [itex]k\in\mathbb{Z}[/itex]? Can [itex]x\in[0,\infty)[/itex]?
 
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  • #2
LCKurtz
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Compute the sine coefficients for [itex]f(x)=e^{-x^{2}}[/itex] on the interval [itex][0,2\pi][/itex]. Does this mean [itex]f(x+2\pi k)=f(x)[/itex], [itex]k\in\mathbb{Z}[/itex]? Can [itex]x\in[0,\infty)[/itex]?

You have to know what periodic function you are expanding in a FS before you can calculate the Fourier coefficients. Do you mean to calculate the coefficients for the odd periodic extension of ##f(x)##? Do you know what the graph of the odd periodic extension would look like? Does that answer your last question?
 
  • #3
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It says on the interval [itex][0,2\pi][/itex]. Does this mean [itex]f(x+2\pi k)=f(x)[/itex], [itex]k\in\mathbb{Z}[/itex]?

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. A real-valued function [itex]f(x)[/itex] of a real variable is called periodic of period [itex]T>0[/itex] if [itex]f(x+T) = f(x)[/itex] for all [itex]x\in\mathbb{R}[/itex].

So, can [itex]x\in[0,\infty)[/itex]?
 
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LCKurtz
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It says on the interval [itex][0,2\pi][/itex]. Does this mean [itex]f(x+2\pi k)=f(x)[/itex], [itex]k\in\mathbb{Z}[/itex]?

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. A real-valued function [itex]f(x)[/itex] of a real variable is called periodic of period [itex]T>0[/itex] if [itex]f(x+T) = f(x)[/itex] for all [itex]x\in\mathbb{R}[/itex].

So, can [itex]x\in[0,\infty)[/itex]?

I think you have answered your own question. Of course if a function is defined on ##[0,2\pi]## and periodic with period ##2\pi## it is defined for all x. But if you want to calculate the coefficients for your f(x) you need to know what periodic extension you are using. That's why I asked you if you mean to use the odd periodic extension of ##e^{-x^2}## to calculate the coefficients.
 
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I need to use the the half-range sine expansion. Correct?
 
  • #6
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I think you have answered your own question. Of course if a function is defined on ##[0,2\pi]## and periodic with period ##2\pi## it is defined for all x. But if you want to calculate the coefficients for your f(x) you need to know what periodic extension you are using. That's why I asked you if you mean to use the odd periodic extension of ##e^{-x^2}## to calculate the coefficients.

I need to use the the half-range sine expansion. Correct? However, the problem does not state that the function is periodic, nor that it is defined on [itex][0,2\pi][/itex]. After all, the Gaussian function is not periodic. My instructor said that I should only consider [itex][0,\infty)[/itex], but then this would not satisfy the definition of a periodic function.
 
  • #7
LCKurtz
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I need to use the the half-range sine expansion. Correct? However, the problem does not state that the function is periodic, nor that it is defined on [itex][0,2\pi][/itex]. After all, the Gaussian function is not periodic. My instructor said that I should only consider [itex][0,\infty)[/itex], but then this would not satisfy the definition of a periodic function.

That's right. But if you are going to find a FS that represents ##e^{-x^2}## on some interval, you must decide what interval. You must know that to use the appropriate half range formulas. I would suggest you ask your instructor what interval he wants. It can't be ##(0,\infty)## unless you are talking about a Fourier transform, not a FS.
 
  • #8
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In this case [itex]T=\pi[/itex], but he said that I should work on the positive x axis only. That's what bothers me. You can't do that, right? [itex]f(x)[/itex] would not be periodic and it would not work because a Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.
 
  • #9
LCKurtz
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In this case [itex]T=\pi[/itex], but he said that I should work on the positive x axis only. That's what bothers me. You can't do that, right? [itex]f(x)[/itex] would not be periodic and it would not work because a Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.

That's right. It isn't going to work on the positive x axis. The half range sine series will only converge to ##e^{-x^2}## on ##(0,\pi)##. What the FS will converge to outside of that interval is the odd ##2\pi## periodic extension of ##e^{-x^2}##, except for multiples of ##\pi##. Your teacher may be referring to the fact that the formula for the coefficients only uses positive values of ##x##.
 

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