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let f(x) be the characteristic function of the interval [a,b] $\subset$ $[-\pi , \pi ]$, show that the Fourier series of $f$ is given by $$ \frac{b-a}{2\pi} + \sum_ {n \neq 0} \frac{e^{-ina} - e^{-inb}}{2\pi i n} e^{inx} $$ The Formula for the Fourier series of the Characteristic function

I have calculated $\hat{f}(0)$ and it was $\frac{b-a}{2\pi}$ as required. and I have calculated $\hat{f}(n)$ and it was equal to $$\hat{f}(n) = \frac{1}{2in\pi} (e^{-ina} - e^{-inb})$$.

So I solved letter(a) of the question it was so easy, thank u very much for anyone helped me. But the remaining part of the question is as follows:

(b)Show that if $a \ne -\pi$ or $b \ne \pi$ and $a\ne b$ then the Fourier Series does not converge absolutely to for any x.

There is a hint in the book for proving this part, but even the application for the hint is not clear for me.

The Hint is:[It suffices to prove that for many values of n one has $|sin n\theta_0| \ge c > 0$ where $\theta_0 = \frac{(b-a)}{2}$]

Could anyone help me?

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    The link is broken. Please format the problem and repost it. Also, you should show what you have tried. – 2017-02-27
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    the link open with me ......but okay I will try to send the question in an obvious way thanks. – 2017-02-27

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In general, the Fourier series of a function with period $2 \pi$ can be given by

$$f(\theta) = \frac{a_0}{2} + \sum_{n=1}^{\infty}(a_n cos(n \theta)+b_n sin(n \theta))$$

Where $$a_n=\frac{1}{\pi} \int_{-\pi}^{\pi} f(\theta) cos n\theta d\theta$$ and $$b_n=\frac{1}{\pi} \int_{-\pi}^{\pi} f(\theta) sin n\theta d\theta$$.

You may also express the fourier series as:

$$f(\theta) = \sum_{-\infty}^{\infty}c_n e^{i n \theta}$$ where

$$c_n=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(\theta) e^{-i n \theta} d \theta$$

Can you, with this information and using $f(\theta)$ as the characteristic function, prove whate they are asking?

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    yes I`ll do and if I have problems I will ask u if u do not mind, thanks. – 2017-02-27