I am dealing with function written in a 2D Legendre polynomial basis and I need to convert it so that it's written in a 2D monomial basis. I've found of an algorithm that allows for change of basis when dealing with univariate polynomials (here: Basis conversions among univariate polynomial representations). However, I can't seem to find a method/means to extend this to work with multivariate polynomial bases. Do there exist algorithms or formulations of conversions from one 2D (or even general multivariate) polynomial basis to another 2D (or general multivariate) basis?
3
-
$\begingroup$ If your 2D Legendre polynomial basis is given by $P_{mn}(x,y) = P_m(x) P_n(y)$, where $P_m$ are the 1D Legendre polynomials, why not just apply your algorithm first in $x$ and then in $y$? $\endgroup$– Igor KhavkineCommented Jul 6, 2023 at 16:28
-
$\begingroup$ The algorithm is written to transform coefficients of a univariate polynomial basis to another. So it's still ambiguous regarding how the algorithm would transform a coefficient for basis function $P_{mn}$ $\endgroup$– David G.Commented Jul 9, 2023 at 2:21
-
1$\begingroup$ If $P_m(x) = \sum_k \Delta_{mk} x^k$, then $P_{mn}(x,y) = \sum_k \sum_l \Delta_{mk} \Delta_{nl} x^k y^l$. You can do the $k$- and $l$-summations in either order. This turns a 1D change of basis into a 2D change of basis. $\endgroup$– Igor KhavkineCommented Jul 9, 2023 at 22:38
Add a comment
|