Is a function space [tex]\{f|f:X\times Y\to \mathbb{R}\}[/tex] a tensor product of spaces [tex]\{f|f:X\to\mathbb{R}\}[/tex] and [tex]\{f|f:Y\to\mathbb{R}\}[/tex]?(adsbygoogle = window.adsbygoogle || []).push({});

Or a more conrete question, which is the one I'm mostly interested, is that if [tex]\{\sin(kx),\cos(kx)\}[/tex] is a basis for one dimensional continuous functions, is [tex]\{\sin(k_1x_1)\sin(k_2x_2),\sin(k_1x_1)\cos(k_2x_2),\cos(k_1x_1)\sin(k_2x_2),\cos(k_1x_1)\cos(k_2x_2)\}[/tex] then basis for two dimensional continuous functions?

Okey I know fourier series can converge towards non-continuous functions, but I mean that if I have an arbitrary continuous two dimensional function, will those combinations of sines and cosines suffice?

btw. I don't know much of fourier series yet. I've learned something on phycisists' method courses. I can see myself that my questions are not fully precise, but I don't know how to make them such.

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# Multidimensional fourier series

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