# Multidimensional fourier series

1. May 18, 2007

### jostpuur

Is a function space $$\{f|f:X\times Y\to \mathbb{R}\}$$ a tensor product of spaces $$\{f|f:X\to\mathbb{R}\}$$ and $$\{f|f:Y\to\mathbb{R}\}$$?

Or a more conrete question, which is the one I'm mostly interested, is that if $$\{\sin(kx),\cos(kx)\}$$ is a basis for one dimensional continuous functions, is $$\{\sin(k_1x_1)\sin(k_2x_2),\sin(k_1x_1)\cos(k_2x_2),\cos(k_1x_1)\sin(k_2x_2),\cos(k_1x_1)\cos(k_2x_2)\}$$ 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.

Last edited: May 18, 2007