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If $f(x)$ is a piecewise continous function in $[-l,l]$ how can we show that its indefinite integral $F(x) = \int _{-l}^x f(s) ds$ has a full Fourier series that converges pointwise?

And, how can we write this convergent series for $F(x)$ explicitly in terms of the Fourier coefficients $a_n, b_n$ of $f(x)$ if $a_0 = 0$?

Someting that came to mind is to applying a convergence theorem.

I really need help with the showing pointwise convergence. the proof below is good but is missing this and I cannot understand why we are dealing with pi instead of l in the proof below

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You can look here for a proof: math.psu.edu/wysocki/M412/Notes412_8.pdf –  Siminore Oct 26 '12 at 9:43
    
I looked through it but cannot filter that info to help me out with this proof. Do you mind using that pdf and placing it here and explaining it further to how it relates to this problem? Then I can ask some follow up question –  mary Oct 26 '12 at 9:56

1 Answer 1

up vote 3 down vote accepted

Theorem. Assume that $f$ is a piecewise continuous function on $[-\pi,\pi]$. For every $x_0$, $x\in [-\pi,\pi]$ there results $$ \int_{x_0}^x f(t)\, dt = \int_{x_0}^x \frac{a_0}{2} + \sum_{n=1}^\infty \int_{x_0}^x \left( a_n \cos nt + b_n \sin nt \right)\, dt. $$ Given $x_0$, the right-hand side converges uniformly in $[-\pi,\pi]$.

Proof. The function $$ F(x)=\int_{x_0}^x \left( f(t)-\frac{a_0}{2} \right)\, dt, \quad x \in [-\pi,\pi] $$ is continous, vanishes at $-\pi$ and at $\pi$, and we can extend it as a $2\pi$-periodic function. Moreover its derivative is continuous except at a finite number of points. By a well-know result, its Fourier series converges uniformly. If $A_n$ and $B_n$ are its Fourier coefficients, then $$ a_n = n B_n, \quad b_n = -n A_n. $$ Hence $$ \begin{align} \int_{x_0}^x \left( f(t)- \frac{a_0}{2} \right)dt &= F(x)-F(x_0) =\\ &= \sum_{n=1}^\infty \left\{ A_n (\cos nx -\cos n x_0 ) + B_n (\sin nx - \sin n x_0 ) \right\}\\ &= \sum_{n=1}^\infty \left\{ -b_n \frac{\cos nx - \cos n x_0}{n}+a_n \frac{\sin nx - \sin nx_0}{n} \right\} \\ &=\sum_{n=1}^\infty \left\{ b_n \int_{x_0}^x \sin nt \, dt + a_n \int_{x_0}^x \cos nt \, dt \right\}. \end{align} $$

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Where is l? I see it replaced by pi, and then the question theorem didn't help –  mary Oct 26 '12 at 18:51
    
We can't just say that its fourier series converges uniformly. Can you please show this? –  mary Oct 27 '12 at 7:38
    
@mary Who cares about $l$? You can always map $[-l,l]$ to $[-\pi,\pi]$ in a bijective way.I guess you should buy a good book in advanced calculus, since continuous piecewise $C^1$ functions can always be expanded in Fourier series, and the convergence is uniform. –  Siminore Oct 27 '12 at 10:12
    
We actually need to invoke convergence in our proof in this problem. While your statement is true, it is missing in the proof –  mary Oct 28 '12 at 2:05

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