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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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    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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