# Bivariate Function Approximation

I am working on a nonlinear control design and having difficulty in finding approximation of bivariate functions. Are there papers or methods discussing the following question:

For any bivariate function F(x,y), find functions G(.), m(x), n(y), such that G(m(x)+n(y)) is best approximation of F(x,y).

Thanks for helping! Wang Tao

• Sorry, I don't know the answer. However, here's something off the top of my head. Any function $H(x,y) = G(m(x)+n(y))$ satisfies the identity $\partial_x \partial_y \log[\partial_x H(x,y)/\partial_y H(x,y)] = 0$. Perhaps there are also other identities like this one. The point is that you might have a hope of approximating $F(x,y)$ with such an $H(x,y)$ only if the left hand side of the identity applied to $F(x,y)$ gives you something "small". Is that the case for you? Have you considered approximations like $F(x,y) \sim \sum_i m_i(x) n_i(y)$? You may be able to find more literature on those. – Igor Khavkine Mar 4 '14 at 12:31
• In which norm would you like to approximate, and what's the regularity of your functions? – Pietro Majer Mar 4 '14 at 13:24
• What about n-dimensional Fourier Transform? – rych Mar 6 '14 at 6:23
• What are the properties of function F? convex or concave? continuous or not? symmetric or not? we need more information to give you an answer. – user39815 Mar 6 '14 at 6:33
• Perhaps this paper will be of interest to you: sciencedirect.com/science/article/pii/0377042794901791 – Shamisen Jan 6 '15 at 15:23