Separation of function When can a function of 2 variables say $h(x,y)$ can be written as $$\sum_i f_i(x)g_i(y)$$  I want to know what conditions on $h$ would ensure this kind of separation. 

If you don't require the sum to be finite then essentially anything expandable in a two dimensional Fourier Series will satisfy what you want. For example if $f(x,y)$ is defined on the unit square, then
$$f(x,y)=\sum_{m,n\in\mathbb{Z}}a_{m,n}e^{2\pi i m x}e^{2\pi i n y}$$ 
for appropriate coefficients $a_{m,n}$. For example, if $f(x,y)$ is continuously differentiable, then the series converges uniformly. Otherwise, you have almost sure convergence for any $L^2$ function by Carleson's theorem.
