Intervals where the function is similar to the Fourier series

Asked May 06 '13 at 10:18

Active May 06 '13 at 23:16

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$$f(x)=\left\{\begin{array}{l l} 0,\quad x \in [-L,0[\\ 1,\quad x \in [0,L] \end{array}\right.$$

I need to know in which intervals the sum of the Fourier series is "equal to the function $f(x)$".

Please help and help me understand how to solve it because I have four exercises like this... Thanks

edited May 06 '13 at 23:16

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asked May 06 '13 at 10:18

Deiota

What convergence theorems for Fourier series do you know? – mrf May 06 '13 at 10:19
A related problem. – Mhenni Benghorbal May 06 '13 at 10:38
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Note that, your function is piecewise smooth. – Mhenni Benghorbal May 06 '13 at 10:45
Yeah that I know. But what about $$x=0$$? Discontinuity? This function has the value $$\frac{1}{2}=\frac{f(0+)+f(0-)}{2}$$ – Deiota May 06 '13 at 11:01
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Consider the convergence on the sub-intervals and at the point of discontinuity, the series coverges to $ \frac{f(x+)+f(x-)}{2} $. You can see this directly if you subs $x=0$ in the series which gives you the value $1/2$. – Mhenni Benghorbal May 06 '13 at 11:03
By the way, do not forget to up vote the answer if you benefited from it. – Mhenni Benghorbal May 06 '13 at 11:07
So I say since the function is piecewise smooth in the intervals $$]-\pi,0[$$ and $$]0,\pi[$$ , at this intervals the function converge to the Fourier series, at 0, $$\pi$$ and $$-\pi$$ the funtion converge to 1/2. Is that it? – Deiota May 06 '13 at 11:07
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You say the series converges uniformly to $f(x)$ on these sub-intervals. – Mhenni Benghorbal May 06 '13 at 11:15
Without the 0, pi and -pi. Don't refer to this points? – Deiota May 06 '13 at 11:17
These intervals $(-\pi,0)$ and $(0,\pi)$. – Mhenni Benghorbal May 06 '13 at 11:21
Note that the values of the series at $-\pi$ and $\pi$ are the same as at $0$ AND YOU KNOW WHY? – Mhenni Benghorbal May 06 '13 at 11:23
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Yes I do :) lol – Deiota May 06 '13 at 11:51
Solved thanks a lot:) – Deiota May 06 '13 at 14:32

Intervals where the function is similar to the Fourier series

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