Christoffel-weighted least-squares polynomial collocation

Let $\mathrm{\Lambda }$ be finite, contain $0$, and let $\psi =({\psi }_{\alpha }{)}_{\alpha \in \mathrm{\Lambda }}$ be the real orthonormal basis of Finite orthonormal polynomial tensor basis. Set $N=\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }$ and $K_{\mathrm{\Lambda }}(y)={\sum }_{\alpha \in \mathrm{\Lambda }}\mathrm{\mid }{\psi }_{\alpha }(y){\mathrm{\mid }}^2\mathrm{.}$ The algorithm takes an evaluable real function $u$, a positive integer sample count $M$, and independent samples $X_1,\dots ,X_M$ from the probability law $d\nu =(K_{\mathrm{\Lambda }}\mathrm{/}N)d\rho$. It forms $G=\frac{1}{M}{\sum }_{i=1}^M\frac{N}{K_{\mathrm{\Lambda }}(X_i)}\psi (X_i)\psi (X_i{)}^T,b=\frac{1}{M}{\sum }_{i=1}^M\frac{N}{K_{\mathrm{\Lambda }}(X_i)}u(X_i)\psi (X_i)\mathrm{.}$ If every eigenvalue of $G$ lies in $[1\mathrm{/}2,3\mathrm{/}2]$, return the unique coefficient vector $c=G^{-1}b$ and polynomial $\widehat{u}={\sum }_{\alpha \in \mathrm{\Lambda }}{c}_{\alpha }{\psi }_{\alpha }$. Otherwise return unstable-sample and no coefficients. The weighted empirical objective minimized on the accepted branch is $\frac{1}{M}{\sum }_{i=1}^M\frac{N}{K_{\mathrm{\Lambda }}(X_i)}\mathrm{\mid }u(X_i)-p(X_i){\mathrm{\mid }}^2,$ over $p\in {\mathcal{P}}_{\mathrm{\Lambda }}$.