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Given a double Fourier series for some $f:[0,L]\times [0,R]\to \mathbb{R}$ of the following form $$\sum_{k,l=0}^{\infty}a_{kl}\cos\left(\frac{2k\pi x}{L}\right)\cos\left(\frac{ly}{R}\right)+b_{kl}\cos\left(\frac{2k\pi x}{L}\right)\sin\left(\frac{ly}{R}\right)+c_{kl}\sin\left(\frac{2k\pi x}{L}\right)\cos\left(\frac{ly}{R}\right)+d_{kl}\sin\left(\frac{2k\pi x}{L}\right)\sin\left(\frac{ly}{R}\right).$$ where $L,R$ are positive numbers and $a_{kl},b_{kl},c_{kl},d_{kl}$ are real coefficients. I wonder whether there exists any function $g$ over $[0,L]\times [0,R]$ which cannot be represented by a series of the above form?

It will be more illustrative if someone can provide a concrete example which is continuous or/and differentiable. I am not looking for some too "crazy" examples which just right in their own sake.

user31899
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  • Are there any restrictions on $g$ (continuity, differentiability, etc.)? If not, functions like $g(x,y) = 1$ if $x,y$ are both rational and $0$ otherwise would be examples. – JimmyK4542 Dec 26 '14 at 00:19
  • Yeah, are there any continuous example, or maybe differentiable example? – user31899 Dec 26 '14 at 01:48
  • http://math.stackexchange.com/questions/14855/an-example-of-a-continuous-function-whose-fourier-series-diverges-at-a-dense-set

    This happens even in one dimension.

     – Ray Yang Jan 02 '15 at 11:59

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