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?