What is the Fourier Series for f(x) = (pi - |x|)^2 on [-pi,pi]?

In summary, the conversation discusses a problem involving Fourier series and the attempt to prove that ∑1/n^4 = (pi^4)/90 using the series and the fact that ∑1/n^2 = (pi^2)/6. The conversation also mentions a mistake in the problem given in the book and the use of Parseval's Theorem to solve the problem with the correct series. The conversation also addresses the discrepancy between the results obtained by the book and the results obtained by the individual.
  • #1
Poopsilon
294
1

Homework Statement



For those of you who own baby rudin this is problem #14 in Chapter 8. For those of you don't I am given that f(x) = (pi - |x|)^2 on [-pi,pi], I need to show that f(x) = (pi^2)/3 + ∑(4/n^2)cos(nx)

The series above is an infinite series from n=1 to infinity. I know I know I need to take an hour or two and learn Latex, I will next time.


Homework Equations



Um, I can't really think of anything, if you are providing help I'm sure you know how to calculate a Fourier series better than I, considering this is my first time.


The Attempt at a Solution



Well I've calculated the Fourier coefficients and checked over the integrals I did to obtain them what now feels like about fifty times. My a_0 is coming out to (pi^2)/3 which is good. As for a_n I get:

a_n = -(4cos(n*pi))/n^2 + 4/n^2.

For b_n I get:

b_n = (4pi*cos(n*pi))/n

Now with these coefficients I am able to finagle my Fourier series into what I want except I keep getting this nasty extra infinite series left over, now I was thinking well ok maybe it is equal to zero, you know telescopes or something, but I plugged in x=0 and it almost certainly isn't. Please any help would be greatly appreciated I am very frustrated, thanks.
 
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  • #2
Hard to say where your errors are. But you might notice that your f(x) is an even function. Your formula for bn should be

[tex]b_n = \frac 1 \pi \int_{-\pi}^{\pi}f(x) \sin(nx)\, dx[/tex]

The integrand is an odd function. What do you get if you integrate an odd function over an interval symmetric about 0?

Also, in computing your an, you will have an even integrand so you can integrate for formula from 0 to pi and double it, thus relieving yourself of the absolute value signs.

Try that. It might help you cut down the mistakes, whatever they are.
 
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  • #3
Oh shoot ok I did the absolute value integral trick for both sine and cosine, but your saying I can only do it for cosine and that I should be getting 0 for the b_n coefficient. So how do I take care of absolute values with odd functions? Do I just drop the absolute value sign without changing the limits of integration and doubling the integral?

Ok So I have my b_n term equal to zero but my I still have my a_n term and so I still have this extra infinite series:

∑ -(4/n^2)cos(n*pi)cos(n*x).

Now if I plug in x=0 here I get ∑ (-4/n^2)cos(n*pi). Now the cos(n*pi) becomes (-1)^n-1, and this pretty clearly does not sum to zero. So I'm still a bit confused.
 
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  • #4
Poopsilon said:
Oh shoot ok I did the absolute value integral trick for both sine and cosine, but your saying I can only do it for cosine and that I should be getting 0 for the b_n coefficient. So how do I take care of absolute values with odd functions? Do I just drop the absolute value sign without changing the limits of integration and doubling the integral? Thanks for your help.

You don't have to integrate the odd function at all. There is a general theorem about symmetry:

[tex]\hbox{If }f(x) = f(-x)\hbox{ then } \int_{-a}^a f(x)\, dx = 2\int_0^a f(x)\, dx[/tex]

and

[tex]\hbox{If }f(x) = -f(-x)\hbox{ then } \int_{-a}^a f(x)\, dx = 0[/tex]

You prove both of them by breaking the integral into two parts, from -a to 0 and 0 to a and letting u = -x on the first one. You can also look at half range Fourier expansions which talk about this symmetry.
 
  • #5
Ah ok thanks, and don't worry about the second part of my question I figured it out.
 
  • #6
Ok so now I have hit a new brick wall on the same problem. I am trying to prove that ∑1/n^4 = (pi^4)/90. Using both the Fourier series result as well as the fact that ∑1/n^2 = (pi^2)/6. I've spent a good portion of my day on this problem to no avail I fear it may involve a better understanding of Fourier series than I currently have, thanks for your help!

Ok update on this, I talked to my TA and he says there is a mistake in the in the problem in the book, it should be f(x) = (pi^2)/3 + ∑(2/n^2)cos(nx), (changing the 4 in the numerator into a 2). So I used Parseval's Theorem to successfully solve ∑1/n^4 = (pi^4)/90 with this change, but this means the first two parts are wrong, which is now driving me crazy because I calculated the Fourier series and got what the book has! And moreover by simply plugging x=0 into (pi - |x|)^2 = (pi^2)/3 + ∑(4/n^2)cos(nx) I got ∑1/n^2 = (pi^2)/6. But now that we have changed the 4 into a 2 I am getting ∑1/n^2 = (pi^2)/3 ! What does this mean? Thanks.
 
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  • #7
Poopsilon said:
Ok so now I have hit a new brick wall on the same problem. I am trying to prove that ∑1/n^4 = (pi^4)/90. Using both the Fourier series result as well as the fact that ∑1/n^2 = (pi^2)/6. I've spent a good portion of my day on this problem to no avail I fear it may involve a better understanding of Fourier series than I currently have, thanks for your help!

Ok update on this, I talked to my TA and he says there is a mistake in the in the problem in the book, it should be f(x) = (pi^2)/3 + ∑(2/n^2)cos(nx), (changing the 4 in the numerator into a 2). So I used Parseval's Theorem to successfully solve ∑1/n^4 = (pi^4)/90 with this change, but this means the first two parts are wrong, which is now driving me crazy because I calculated the Fourier series and got what the book has! And moreover by simply plugging x=0 into (pi - |x|)^2 = (pi^2)/3 + ∑(4/n^2)cos(nx) I got ∑1/n^2 = (pi^2)/6. But now that we have changed the 4 into a 2 I am getting ∑1/n^2 = (pi^2)/3 ! What does this mean? Thanks.

The series with the 4 is the correct FS for the given function. There is no doubt about that and, as you have observed, it gives the correct sum for ∑1/n2. I also plotted several terms of this FS and it does converge to the function. So it is correct.

Something strange going on here and I don't see what it is. Likely something simple we are both overlooking. I haven't looked at Parseval's Theorem for many years. Are you sure you are applying it correctly?
 
  • #8
Ok just to wrap this thread up my TA was wrong. It is supposed to be a 4, so the first two parts are correct. I was applying Parseval's incorrectly, it requires the infinite series in the form which involves e^inx and in that form it is 2/n^2 and so everything works out correctly. Thanks for your help.
 

FAQ: What is the Fourier Series for f(x) = (pi - |x|)^2 on [-pi,pi]?

What is the Rudin Fourier Series Problem?

The Rudin Fourier Series Problem is a mathematical problem named after the mathematician Walter Rudin. It asks whether every real-valued continuous function on the unit circle can be represented as a Fourier series with respect to the standard orthonormal basis.

What makes the Rudin Fourier Series Problem significant?

The Rudin Fourier Series Problem is significant because it addresses the fundamental question of whether every continuous function can be represented as a Fourier series. It also has connections to several other areas of mathematics, such as harmonic analysis and complex analysis.

Has the Rudin Fourier Series Problem been solved?

No, the Rudin Fourier Series Problem remains an open problem in mathematics. There have been partial results and progress made in understanding the problem, but a complete solution has not yet been found.

What are some related problems to the Rudin Fourier Series Problem?

Some related problems include the question of whether every continuous function can be approximated by trigonometric polynomials, and whether every continuous function can be represented as a Fourier series with respect to a different orthonormal basis, such as the Haar basis.

How does the solution to the Rudin Fourier Series Problem impact other areas of mathematics?

The solution to the Rudin Fourier Series Problem would have significant implications for other areas of mathematics, such as harmonic analysis, complex analysis, and functional analysis. It would also contribute to a deeper understanding of the structure and properties of continuous functions and their representations.

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