Compute Sin Coeffs for f(x)=e^(-x^2) on [0,2π]

[glebovg]

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

 

[LCKurtz]

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

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?

 

[glebovg]

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

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

So, can x\in[0,\infty)?

 

[LCKurtz]

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

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

So, can x\in[0,\infty)?

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.

 

[glebovg]

I need to use the the half-range sine expansion. Correct?

 

[glebovg]

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 [0,2\pi]. After all, the Gaussian function is not periodic. My instructor said that I should only consider [0,\infty), but then this would not satisfy the definition of a periodic function.

 

[LCKurtz]

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 [0,2\pi]. After all, the Gaussian function is not periodic. My instructor said that I should only consider [0,\infty), 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.

 

[glebovg]

In this case T=\pi, but he said that I should work on the positive x-axis only. That's what bothers me. You can't do that, right? f(x) 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.

 

[LCKurtz]

In this case T=\pi, but he said that I should work on the positive x-axis only. That's what bothers me. You can't do that, right? f(x) 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 $x$.