Does Fourier series of x^2 converge?

In summary, the conversation discusses the convergence of the Fourier series for the function f(x)=x^2 on the interval [-\pi,\pi]. The speaker is trying to show the convergence and asks for help, mentioning their attempt using Dirichlet's test. They also mention wanting to show uniform convergence, and another speaker suggests using the Weierstrass M-test. Another speaker mentions a result from "Fourier Analysis" by Stein and Shakarchi that states if the Fourier series of a continuous function on the circle is absolutely convergent, it will converge uniformly to the function. The conversation ends with the group discussing this result.
  • #1
gauss mouse
28
0
I'm trying to show that the Fourier series of [itex] f(x)=x^2[/itex] converges and I can't. Does anybody know if it actually does converge? (I'm assuming that [itex] f(x)=x^2[/itex] for [itex] x\in [-\pi,\pi][/itex]).
The Fourier Series itself is [itex] \displaystyle\frac{\pi^2}{3}+4\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\cos nx [/itex]
I tried using Dirichlet's test but it wasn't working for me, though that may be because I'm doing something wrong.
 
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  • #2
Isn't it easy to show absolute convergence?? Since [itex]\sum \frac{1}{n^2}[/itex] converges

Do you also want to show that it converges to [itex]x^2[/itex]?? There are many theorems out there that give you that, so it depends on what you have seen.
 
  • #3
I was just looking to see if it converged. So yes, you're absolutely right. I was going well off-beam with my attempt.

And about converging to [itex]x^2[/itex]; I guess it does so uniformly since the function is continuous on the circle and the Fourier series converges absolutely.

Thank you!
 
  • #4
gauss mouse said:
I was just looking to see if it converged. So yes, you're absolutely right. I was going well off-beam with my attempt.

And about converging to [itex]x^2[/itex]; I guess it does so uniformly since the function is continuous on the circle and the Fourier series converges absolutely.

Thank you!

It's not because a function is continuous and because the Fourier series converges absolutely, that you can have uniform convergence (I think).
Here, I think you can infer uniform convergence from the Weierstrass M-test.
 
  • #5
No I think it is. I quote Corollary 2.3 from "Fourier Analysis" by Stein and Shakarchi -

"Suppose that [itex]f [/itex] is a continuous function on the circle and that the Fourier series of [itex]f [/itex] is absolutely convergent, [itex]\sum_{n=-\infty}^\infty |\hat{f}(n)|<\infty. [/itex] Then, the Fourier series converges uniformly to [itex]f[/itex], that is
[itex]\displaystyle \lim_{N\to\infty}S_N(f)(\theta)=f(\theta) [/itex] uniformly in [itex]\theta. [/itex]
 
  • #6
Oh ok, I did not know that result. Nice!
 
  • #7
Yeah it's pretty sweet. It's not too restrictive.
 

1. What is the Fourier series of x^2 and does it converge?

The Fourier series of x^2 is given by , and it converges to x^2 for all real values of x. This means that the Fourier series of x^2 is a valid representation of the function x^2.

2. How is the convergence of a Fourier series determined?

The convergence of a Fourier series is determined by the coefficients of the series and the properties of the function being represented. In general, if a function is continuous and has a finite number of discontinuities, the Fourier series will converge to the function. However, for functions with more complex properties, the convergence may need to be tested using specific convergence tests.

3. Can a Fourier series diverge?

Yes, a Fourier series can diverge. This occurs when the function being represented is not continuous or has an infinite number of discontinuities. In these cases, the Fourier series may not be a valid representation of the function, and the series may not converge.

4. How is the Fourier series of x^2 different from the Fourier series of other functions?

The Fourier series of x^2 is unique in that it includes only cosine terms and no sine terms. This is because the function x^2 is an even function, meaning it is symmetric about the y-axis, and thus only cosine terms are needed to represent it. Other functions may have a combination of cosine and sine terms in their Fourier series, depending on their symmetry properties.

5. Can the Fourier series of x^2 be used to approximate other functions?

Yes, the Fourier series of x^2 can be used to approximate other even functions. This is because the Fourier series of x^2 is a general representation of all even functions, and thus can be used to approximate any other even function. However, for odd functions, a combination of cosine and sine terms would be needed for an accurate approximation.

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