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I want to find out whether the Fourier series of $\partial_x^i f$ and $\partial_y^i f$ converges absolutely if $f$ is a function in $L^2$ and both of its fourth partial derivatives exist and are continuous.

First in the case of $i = 0$ I am trying to compute

$$ a_0 = \frac{1}{\pi} \int_{-\pi}^\pi f(x,y) dx = F(y) \in L^2$$

$$ a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x,y) \cos(nx) dx $$

$$ b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x,y) \sin(nx) dx $$

I've tried integration by parts on $a_n$ but it doesn't work and I don't see how integration by substitution could do any good. Then I thought maybe I can argue like this:

$$ a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x,y) \cos(nx) dx \leq \frac{1}{\pi} \int_{-\pi}^\pi f(x,y) dx $$

and then

$$ \sum | a_n | \leq \frac{1}{\pi} |F(y)| \sum 1$$

but that doesn't help either. What other ways are there to compute an integral? Many thanks for your help.

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    Integrating by parts is a good idea. Why didn't it work?2011-11-02
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    I ended up having $a_n = K + a_n$. Maybe I made a mistake, although I did it twice...2011-11-02
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    You can take a primitive of $\cos (nx)$ and take the derivative of $f$ with respect to $x$, then do a similar thing for the integral which will appear.2011-11-02

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