I would like to approximate some positive, scalar function, $f(x,y) > 0$, on a 2D field of finite size i.e. $x=[a,b],y=[c,d]$
I am familiar with the set of basis functions used in the Fourier series but using the Fourier series requires too many terms. Infinitely many:)
Instead, I desire to find a collection of functions that can be used to approximate $f(x,y)$ with a "finite" number of terms.
How can I approximate $f(x,y)$ with a collection of functions $G(x)$ and $H(y)$ (see definition of $G$ and $H$ below), such that the number of elements in $G$ or $H$ is finite?
$$G(x) = [g_1(x)\ldots g_n(x)]\\ H(y) = [h_1(y)\ldots h_n(y)]$$
The elements in $G$ and $H$ do not have to be of the same form ($g_1(x)=\sin(nx),g_2(x)=\sin(mx)$)
Thanks in advance:)
I should clarify that $G(x)$ and $H(y)$ need to be combined in an inner product fashion i.e.
$$f(x,y) = < G(x),H(y) > = g_1(x)h_1(y) + g_2(x)h_2(y) + ... g_n(x)h_n(y)$$