Pointwise convergence of integral of Fourier series

In summary, the problem is to show that for a piecewise-continuous function f(x) in [-L,L], its indefinite integral F(x) = \int_{-L}^x f(s) ds has a full Fourier series that converges pointwise. This means that the series \sum_{n=1}^\infty \frac{1}{2}A_0 + A_n \cos (\frac{n \pi }{L}x) + B_n \sin (\frac{n \pi}{L}x) converges to F(x) at each point in (-L,L). To justify integrating term-by-term, we need uniform convergence. Further work is needed to determine if this is the solution.
  • #1
twizzy
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Homework Statement


If [tex]f(x)[/tex] is a piecewise-continuous function in [tex][-L,L][/tex], show that its indefinite integral [tex]F(x) = \int_{-L}^x f(s) ds[/tex] has a full Fourier series that converges pointwise.

Homework Equations


Full Fourier series: [tex]f(x)=\frac{1}{2}A_0 + \sum_{n=1}^\infty A_n \cos (\frac{n \pi }{L}x) + B_n \sin (\frac{n \pi}{L}x) [/tex]

Definition: [tex]\sum_{n=1}^\infty f_n (x) [/tex] converges to [tex]f(x)[/tex] pointwise in [tex](a,b)[/tex] if for each [tex]a<x<b[/tex] we have
[tex]\Big| f(x) - \displaystyle{\sum_{n=1}^\infty f_n (x)} \Big| \to 0 [/tex] as [tex]N\to\infty[/tex].

The Attempt at a Solution


I think I need to somehow justify integrating term-by-term, but am not sure how to proceed. Any ideas?
 
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  • #2
If you want to integrate term by term, you need uniform convergence.
Haven't really looked at this, so not saying that term by term integration is the solution here.
 

1. What is pointwise convergence of the integral of Fourier series?

Pointwise convergence of the integral of Fourier series refers to the property of a Fourier series where the partial sums of the series converge point by point to the original function. In other words, as the number of terms in the series increases, the value of the integral of the series at each point approaches the value of the original function at that point.

2. How is pointwise convergence of the integral of Fourier series different from uniform convergence?

Pointwise convergence refers to the convergence of the series at each point, whereas uniform convergence refers to the convergence of the series as a whole. In other words, pointwise convergence considers the behavior of the series at individual points, while uniform convergence considers the overall behavior of the series.

3. What is the significance of pointwise convergence of the integral of Fourier series?

The pointwise convergence of the integral of Fourier series is important because it guarantees that the Fourier series can be used to approximate the original function at every point. This is essential in many applications, such as signal processing and data compression.

4. Is pointwise convergence of the integral of Fourier series always guaranteed?

No, pointwise convergence is not always guaranteed. In some cases, the Fourier series may not converge at certain points or may not converge at all. This is dependent on the properties of the original function and the coefficients of the Fourier series.

5. Are there any conditions for pointwise convergence of the integral of Fourier series?

Yes, there are certain conditions that must be met for pointwise convergence of the integral of Fourier series. These conditions include the continuity of the original function and the convergence of the Fourier coefficients to zero as the index increases.

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