Quadrature estimates of polynomial-chaos coefficients

Let $\mathrm{\Lambda }$ be a finite subset of the admissible degree box, with $({\psi }_{\alpha }{)}_{\alpha \in \mathrm{\Lambda }}$ as in Finite orthonormal polynomial tensor basis, and let $Q$ be any declared finite cubature functional $Q(g)={\sum }_{q=1}^J{w}_{q}g(y^{(q)})$ whose nodes lie in the domain of evaluation of $g$. For an evaluable real function $u$, define the quadrature coefficient and the resulting polynomial by ${\widehat{c}}_{\alpha }:=Q(u{\psi }_{\alpha }),{\widehat{P}}_{\mathrm{\Lambda }}u:={\sum }_{\alpha \in \mathrm{\Lambda }}{\widehat{c}}_{\alpha }{\psi }_{\alpha }\mathrm{.}$ When $u$ represents an element of $L^2(\rho )$, its exact coefficients $c_{\alpha }$ and projection $P_{\mathrm{\Lambda }}u$ are those of Finite polynomial-chaos orthogonal projection. The quadrature quantities use the specified node values of $u$ and need not be invariant under changes on null sets. This definition gives no accuracy claim for a general cubature rule.