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

  • $\begingroup$ 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. $\endgroup$ Mar 4 '14 at 12:31
  • 2
    $\begingroup$ In which norm would you like to approximate, and what's the regularity of your functions? $\endgroup$ Mar 4 '14 at 13:24
  • $\begingroup$ What about n-dimensional Fourier Transform? $\endgroup$
    – rych
    Mar 6 '14 at 6:23
  • 1
    $\begingroup$ 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. $\endgroup$
    – user39815
    Mar 6 '14 at 6:33
  • $\begingroup$ Perhaps this paper will be of interest to you: sciencedirect.com/science/article/pii/0377042794901791 $\endgroup$
    – Shamisen
    Jan 6 '15 at 15:23

Expanding on my comment ...

There's some material on bivariate approximation in Carl deBoor's book entitled "A Practical Guide to Splines".

Specifically, chapter XVII has results on approximation by tensor products and code to implement this. The code is also available in the matlab spline package.


I found one book on website, "MULTIVARIATE APPROXIMATION THEORY – SELECTED TOPICS", by E.W. CHENEY. On page 8, there is one equation (the top one of the four equations of that page) that is exactly what I mentioned here but I have difficulty to find more details about it. At end of the book, there are long references but I am unable to figure out which articles are relevant. Can someone highlight me?

Do appreciate your helps and I will update I make any progress.

Best regards. Wang Tao


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