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Let's say we have a complete set of functions ##u_{i} (x)## that can be used to represent any one dimensional function ##f(x)##. We then find another and different set ##v_{i} (x)## that can do the same thing, i.e. represent any function ##f(x)## via a linear superposition.

I believe that any two-dimensional function ##g(x,y)## can then be represented as a linear superposition of weighted products ##u_{i} (x) v_{j} (y)##: $$g(x,y=\Sigma a_{ij} u_{i} (x) v_{j} (y)$$

Is that correct? How do we call this process and when it is feasible?

I know that in Fourier theory a traveling field ##f(x,t)## can be expressed as a weighted sum of traveling plane waves which are not product function of time and space....

Thanks