Smolyak polynomial exactness from actual one-dimensional levels

Let $d\ge 1$ and let $(Q_{\mathrm{\ell }})$ be a nested hierarchy as in Nested one-dimensional quadrature levels and differences. Suppose nondecreasing integers $r_{\mathrm{\ell }}\ge 0$ are supplied such that $Q_{\mathrm{\ell }}(p)={\int }_0^{1}p$ for every real polynomial $p$ of degree at most $r_{\mathrm{\ell }}$. Let $p_j$ be a polynomial of degree at most $s_j$ in coordinate $j$, and suppose $q_j$ is the least level with $s_j\le r_{q_j}$. If a finite downward-closed Smolyak set $\mathrm{\Lambda }$ contains the whole rectangle $R_q=\{\mathrm{\ell }:0\le {\mathrm{\ell }}_j\le q_j\}$, then its rule Smolyak cubature on a finite downward-closed index set exactly integrates every finite linear combination of the tensor polynomials $p_1(x_1)\cdots p_d(x_d)$ with those coordinate-degree bounds. In particular, for ${\mathrm{\Lambda }}_n:=\{\mathrm{\ell }\in {\mathbb{N}}_0^d:\mathrm{\mid }\mathrm{\ell }{\mathrm{\mid }}_1\le n\},$ the rule is exact whenever ${\sum }_j{q}_j\le n$. The $r_{\mathrm{\ell }}$ are the exactness degrees of the actual one-dimensional levels, distinct from the node counts $m_{\mathrm{\ell }}$ in the hierarchy definition; the conclusion does not infer exactness degrees from node counts.

For the selected composite-Simpson hierarchy, each level is exact through degree three, so every tensor polynomial with coordinate degrees at most three is exact on any admissible $\mathrm{\Lambda }$.