# Fourier Series Limit proof

• May 28th 2009, 09:51 AM
TiRune
Fourier Series Limit proof
I'm working on a problem at the moment and I just can't figure it out, here goes:

$\displaystyle \lim_{n \rightarrow \infty } \frac{1}{\pi} \int_{-l}^l f(x) \frac{sin(nx)}{x} dx = f(0)$

with l > 0 and f(x) differentiable on the interval [-l,l]

Things that come to mind are Riemann-Lebesgue Lemma -- from Wolfram MathWorld Riemann's lemma which states that any limit of n to infinity of an integral of a function on -l to l of the function times sin(nx) or cos(nx) tends to 0.
And ofcourse that $\displaystyle \int_0^{\infty}\frac{sin(x)}{x}dx = \frac{\pi}{2}$

Does anyone have any ideas on how I can proceed?
• May 28th 2009, 12:23 PM
Media_Man
Clarity?
Substitute $\displaystyle y=\frac{L}{\pi}x$, so $\displaystyle dx=\frac{\pi}{L}dy$, and sub $\displaystyle g(x)=\frac{f(x)}{x}$

Then $\displaystyle \lim_{n \rightarrow \infty } \frac{1}{\pi} \int_{-l}^l f(x) \frac{sin(nx)}{x} dx$ becomes $\displaystyle \frac{L}{\pi^2} \lim_{n \rightarrow \infty }\int_{-\pi}^\pi g(y)\sin(ny) dy$

So by the Riemann-Lebesgue Lemma, the LHS is zero, so $\displaystyle f(0)=0$

*What exactly is it you have to prove? Are you trying to find a function $\displaystyle f(x)$ that satisfies the equation, or are you supposed to prove the equality holds for all $\displaystyle f(x)$? (Thinking)
• May 28th 2009, 02:49 PM
TiRune
We have to prove that the limit over that integral is equal to f(0) (not zero, but the value f(0) for any f differentiable above -l,l).

Although in the case x is not 0 this proves that the LHS goes to 0, but not that it goes to f(0) :(
• May 28th 2009, 03:13 PM
Sampras
Quote:

Originally Posted by Media_Man
Substitute $\displaystyle y=\frac{L}{\pi}x$, so $\displaystyle dx=\frac{\pi}{L}dy$, and sub $\displaystyle g(x)=\frac{f(x)}{x}$

Then $\displaystyle \lim_{n \rightarrow \infty } \frac{1}{\pi} \int_{-l}^l f(x) \frac{sin(nx)}{x} dx$ becomes $\displaystyle \frac{L}{\pi^2} \lim_{n \rightarrow \infty }\int_{-\pi}^\pi g(y)\sin(ny) dy$

So by the Riemann-Lebesgue Lemma, the LHS is zero, so $\displaystyle f(0)=0$

*What exactly is it you have to prove? Are you trying to find a function $\displaystyle f(x)$ that satisfies the equation, or are you supposed to prove the equality holds for all $\displaystyle f(x)$? (Thinking)

So you basically "transformed" the LHS so that you could use the Riemann-Lebesgue Lemma showing that it is $\displaystyle 0$. But this does not mean its $\displaystyle f(0)$.
• May 28th 2009, 03:45 PM
Sampras
Quote:

Originally Posted by Media_Man
Substitute $\displaystyle y=\frac{L}{\pi}x$, so $\displaystyle dx=\frac{\pi}{L}dy$, and sub $\displaystyle g(x)=\frac{f(x)}{x}$

Then $\displaystyle \lim_{n \rightarrow \infty } \frac{1}{\pi} \int_{-l}^l f(x) \frac{sin(nx)}{x} dx$ becomes $\displaystyle \frac{L}{\pi^2} \lim_{n \rightarrow \infty }\int_{-\pi}^\pi g(y)\sin(ny) dy$

So by the Riemann-Lebesgue Lemma, the LHS is zero, so $\displaystyle f(0)=0$

*What exactly is it you have to prove? Are you trying to find a function $\displaystyle f(x)$ that satisfies the equation, or are you supposed to prove the equality holds for all $\displaystyle f(x)$? (Thinking)

Shouldn't it be $\displaystyle \frac{L}{\pi^2} \lim_{n \to \infty} \int_{-L^2/\pi}^{L^2/\pi} g(y) \sin \left (\frac{n \pi y}{L} \right) \ dy$?
• May 28th 2009, 05:04 PM
Media_Man
Blooper, and yes, f(0)=0
Quote:

Originally Posted by Media_Man
Substitute $\displaystyle y=\frac{L}{\pi}x$, so $\displaystyle dx=\frac{\pi}{L}dy$, and sub $\displaystyle g(x)=\frac{f(x)}{x}$

This was a typo. Sub $\displaystyle y=\frac{\pi}{L}x$ and $\displaystyle dx=\frac{\pi}{L}dy$, in order to change the limits of integration from $\displaystyle [-L,L]$ to $\displaystyle [-\pi,\pi]$ in accordance with the R-L lemma. Thanks for catching that blooper, Sampras. (Itwasntme)

Quote:

But this does not mean its $\displaystyle f(0)=0$.
Yes it does. Completing the RHS of the substitution, $\displaystyle \frac{L}{\pi^2} \lim_{n \rightarrow \infty }\int_{-\pi}^\pi g(y)\sin(ny) dy=f(0)$

Regardless of what f is, it needs to sit on the origin. Here is a proof: If $\displaystyle \int_{-a}^a h(x)dx=0$ for all $\displaystyle a\in\mathbb{R}$, then h must be an even function, that is, symmetric about the y-axis. So, we can think of our integral in question as "tending towards" an even function as n grows large. Obviously, there does not have to be (probably won't be) an n large enough that $\displaystyle h(-x)=h(x)$ but $\displaystyle \lim_{n\rightarrow\infty} h(x)-h(-x)$ does need to be zero.

Consider this counterexample: $\displaystyle f(x)=x+1$, so $\displaystyle f(0)\neq0$ Theorem: $\displaystyle \lim_{n \rightarrow \infty } \frac{1}{\pi} \int_{-l}^l \frac{x+1}x sin(nx) dx\neq0$

Proof: $\displaystyle \lim_{n\rightarrow\infty} \frac{x+1}x sin(nx) - \frac{-x+1}{-x} sin(-nx)$=$\displaystyle \lim_{n\rightarrow\infty} \frac{\sin(nx)}x [(x+1)-(1-x)]$=$\displaystyle \lim_{n\rightarrow\infty} 2\sin(nx)\neq 0$

*I'm not sure how rigorous this is, but because the $\displaystyle \frac{x+1}x sin(nx)$ is not tending to an even function as n gets large, the area bounded under it on either side of the y-axis will always be inequal for some choice of L.
• May 28th 2009, 05:07 PM
Sampras
Quote:

Originally Posted by Media_Man
This was a typo. Sub $\displaystyle y=\frac{\pi}{L}x$ and $\displaystyle dx=\frac{\pi}{L}dy$, in order to change the limits of integration from $\displaystyle [-L,L]$ to $\displaystyle [-\pi,\pi]$ in accordance with the R-L lemma. Thanks for catching that blooper, Sampras. (Itwasntme)

Yes it does. Completing the RHS of the substitution, $\displaystyle \frac{L}{\pi^2} \lim_{n \rightarrow \infty }\int_{-\pi}^\pi g(y)\sin(ny) dy=f(0)$

Regardless of what f is, it needs to sit on the origin. Here is a proof: If $\displaystyle \int_{-a}^a h(x)dx=0$ for all $\displaystyle a\in\mathbb{R}$, then h must be an even function, that is, symmetric about the y-axis. So, we can think of our integral in question as "tending towards" an even function as n grows large. Obviously, there does not have to be (probably won't be) an n large enough that $\displaystyle h(-x)=h(x)$ but $\displaystyle \lim_{n\rightarrow\infty} h(x)-h(-x)$ does need to be zero.

Consider this counterexample: $\displaystyle f(x)=x+1$, so $\displaystyle f(0)\neq0$ Theorem: $\displaystyle \lim_{n \rightarrow \infty } \frac{1}{\pi} \int_{-l}^l \frac{x+1}x sin(nx) dx\neq0$

Proof: $\displaystyle \lim_{n\rightarrow\infty} \frac{x+1}x sin(nx) - \frac{-x+1}{-x} sin(-nx)$=$\displaystyle \lim_{n\rightarrow\infty} \frac{\sin(nx)}x [(x+1)-(1-x)]$=$\displaystyle \lim_{n\rightarrow\infty} 2\sin(nx)\neq 0$

*I'm not sure how rigorous this is, but because the $\displaystyle \frac{x+1}x sin(nx)$ is not tending to an even function as n gets large, the area bounded under it on either side of the y-axis will always be inequal for some choice of L.

$\displaystyle dx = \frac{L}{\pi} dy$.
• May 28th 2009, 05:12 PM
Media_Man
Gahh