I have the following set of basis functions: $x^iy^j$ where $i,j<3$. In other words my basis functions are the bivariate monomials up to third degree. I would like to find an orthonormal or orthogonal polynomial basis. I know that when the monomials are only one variable we can use Gram-Schmidt and obtain an orthonormal basis. However, I have no idea how to proceed when using bivariate monomials. I have read some papers about it but I cannot find any sort of formula or algorithm that would allow me to generate these basis functions.

Does any one know about how to generate these basis functions? Or point me in the direction of relevant literature?

  • $\begingroup$ In order to be able to speak about othogonality or orthonormality ine has to define an inner product on the vector space generated by the vectors $1,x, y, x^2, xy, \ldots, x^2y^2$. Whar inner product did you have in mind? $\endgroup$ – Marc Bogaerts Oct 25 '16 at 17:04
  • $\begingroup$ Well I have found an interesting paper I'm in the process of understanding. They define the following inner product $(f,g) = \int \int W(x,y)f(x,y)g(x,y)dx dy$. With $W$ being a weighting function. $\endgroup$ – RCountZero Oct 25 '16 at 17:27
  • $\begingroup$ So you can apply the Gram-Schmidt algorithm on the vectors $1,x, x^2, y, y^2, xy, \ldots$. $\endgroup$ – Marc Bogaerts Oct 25 '16 at 17:34
  • $\begingroup$ Indeed, it is possible and it works well. I'll post an answer and link to the paper as soon as I can. $\endgroup$ – RCountZero Oct 27 '16 at 8:42

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