I need help to understand how to use the result given by an algorithm for constructing an orthonormal polynomial basis over $L^{2}(X)$, where $X\subset\mathbb{R}^2$, with respect to the inner product $(f,g) = \iint_X f(x,y)g(x,y)\,dxdy$. We denote $\mathcal{P}_{n} = \{\text{Polynomials }P(x,y): \text{ total degree of } P \leq n\}\subset L^{2}(X)$. The algorithm is as follows.
(1) Consider the vector corresponding to the constant polynomial $1$. Normalize it. The resulting vector provides an orthonormal basis of $\mathcal{P}_{0}$.
(2) Suppose we have constructed an orthonormal basis of $\mathcal{P}_{k}$, consisting of $K$ vectors. Multiply each of the vectors in this basis by $x$ and $y$, respectively, and put the resulting vectors, together with the vectors in the basis of $\mathcal{P}_{k}$, in a matrix having $3K$ columns. Calculate the SVD of the matrix. The SVD yields an orthonormal basis of the column space of the matrix, which is also an orthonormal basis of $\mathcal{P}_{k+1}$.
(3) Apply the procedure described in step (2) for each $k$ in $\{0, 1, \ldots , n − 1\}$ in order to obtain an orthonormal basis of $\mathcal{P}_n$.
My thoughts: One need to implement an SVD that uses projection with respect to the given inner product. Also, I assume that a quadrature rule is given to compute the inner product.
Questions: (1) What does it mean to multiply with x and y in step (2)? The SVD is not performed symbolically. Should I sample $X$ and multiply by those values?
(2) Executing the algorithm above, how do I use the result to evaluate the resulting orthonormal polynomial basis at a point in $X$? How do I take the derivative of this basis?
