Smolyak cubature on a finite downward-closed index set

Fix a dimension $d\ge 1$. A finite set $\mathrm{\Lambda }\subseteq {\mathbb{N}}_0^d$ is downward closed if, whenever $\mathrm{\ell }\in \mathrm{\Lambda }$ and ${\mathrm{\ell }}_j>0$, the index $\mathrm{\ell }-e_j$ also belongs to $\mathrm{\Lambda }$; here $e_j$ is the $j$th coordinate unit vector. In this pair an admissible Smolyak set is a nonempty finite downward- closed $\mathrm{\Lambda }$ that contains $0$.

For a level hierarchy from Nested one-dimensional quadrature levels and differences, define the scalar detail value ${\delta }_{\mathrm{\ell }}(f):=({\mathrm{\Delta }}_{{\mathrm{\ell }}_1}^{(1)}\otimes \cdots \otimes {\mathrm{\Delta }}_{{\mathrm{\ell }}_d}^{(d)})(f),$ and the Smolyak cubature functional $Q_{\mathrm{\Lambda }}(f):={\sum }_{\mathrm{\ell }\in \mathrm{\Lambda }}{\delta }_{\mathrm{\ell }}(f)\mathrm{.}$ Each summand is a finite signed weighted sum. Expanding all differences and adding coefficients at equal points gives one finite signed weight for each distinct node in ${\mathcal{N}}_{\mathrm{\Lambda }}:={\bigcup }_{\mathrm{\ell }\in \mathrm{\Lambda }}{X}_{{\mathrm{\ell }}_1}\times \cdots \times X_{{\mathrm{\ell }}_d}\mathrm{.}$ The node-union work count for this direct evaluation convention is $W(\mathrm{\Lambda }):=\mathrm{\mid }{\mathcal{N}}_{\mathrm{\Lambda }}\mathrm{\mid }$, so shared nodes are counted once. A combined weight may be negative or zero; no positivity claim follows from the definition.