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Consider a function $f_p : \mathbb{R} \to \mathbb{R}$ which is continuously differentiable in $(0,2\pi)$ except at two points $x = x_c$ and $x = x_o$. At $x = x_c$, $f_p(x)$ has a jump discontinuity. At the point $x = x_o$, $f_p'(x_o^+)$ and $f_p'(x_o^-)$ exist and are not equal.

Now define a function $f$ equal to $f_p$ on $(0,2\pi)$ and let it be a periodic function with period $2\pi$. Let $\hat{f}_k$ be the Fourier series coefficients of $f$. What I would like to know is whether the Fourier series defined by the coefficients $ik\hat{f}_k$ converge to $$\frac{f'(x_o^+)+f'(x_o^-)}{2}$$ at $x = x_o$ ?

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    $g=f_p'$ is piecewise continuous and the Fourier series of $g$, when evaluated at $x_0$, should converge to the average of its left and right limits at $x_0$. So you are essentially asking if $is\hat{f}(s)=\hat{g}$. But if this is true, we should also have $is\widehat{(f+H)}(s)=\hat{g}$, where $H$ is the Heaviside function, because $f'=(f+H)'$. This implies that $\hat{H}(s)=0$, which is weird. So I guess the answer to your question is negative.2011-09-02
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    @user1551 : you are probably right. I remember reading that it diverges at such points. I don't have the reference book in hand right now (Trig. Series by Zygmund) and hence this question...someone pleas confirm in form of an answer, even without any proof.2011-09-02

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