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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. – user1551 Sep 2 '11 at 6:40
@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. – Rajesh Dachiraju Sep 2 '11 at 8:25

It will diverge, but if I am not mistaken, can be Cesàro summed.

Your function can be written as a sum of a Lipschitz function that is continuously differentiable except at (at most) two points, and a sawtooth wave. The first component has Fourier series that converges, and has the property that the Fourier series of the derivative is $ik\hat{f}_k$. The second component, however, gives the problem. A sawtooth wave has Fourier series expansion (up to a normalising constant) $\hat{f}_k = \frac{(-1)^k}{k}$. So the main issue is handling the sum of the form

$$ \sum (-1)^k\sin(k x) $$

This of course is not absolutely convergent. But away from the singularity, you can take the Cesàro sum: the partial sum of the above expression is more or less the Dirichlet kernel (shifted by a constant). And so the partial Cesàro sum is (roughly) just the Fejér kernel, which is known to converge point-wise away from the singularity.

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It is Cesaro summable $(C,r)$ away from the singularity, only for $r > 1$. (given in the book Trignometric Series by Walker Page no. 55 section 3.5)...but by imposing a condition that the singularity to be that of a jump type, would we be able to make it $(C,1)$ summable ? could there be any such possibility ? – Rajesh Dachiraju Sep 10 '11 at 6:45

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