Smolyak mixed-regularity bounds control chaos coefficient error

Assume $\rho$ is uniform product probability on $[0,1{]}^d$, so its integral is $I_d$ in Explicit work and error rate for dyadic Simpson Smolyak cubature. Let $p$ be an admissible cutoff vector for this law and let $\mathrm{\Lambda }$ be a finite subset of its admissible degree box, with basis from Finite orthonormal polynomial tensor basis. Let $Q_{{\mathrm{\Lambda }}_n}$ be that theorem's dyadic Simpson Smolyak rule, with $n\ge 2d$. Let $u\in L^2(\rho )$ be real and fix a representative $\tilde{u}$ defined at every node of $Q_{{\mathrm{\Lambda }}_n}$. For every $\alpha \in \mathrm{\Lambda }$, suppose the coefficient integrand $g_{\alpha }=\tilde{u}{\psi }_{\alpha }$ has the continuous mixed fourth derivatives and finite bounds $M_{\alpha ,S}$ required by Mixed-derivative bounds for dyadic Simpson details. Define $E_{\alpha ,n}:=C_d\mathrm{\parallel }{g}_{\alpha }{\mathrm{\parallel }}_{\mathrm{m}\mathrm{i}\mathrm{x},d}{W}_{n,d}^{-4}(\log (2+W_{n,d}){)}^{5(d-1)},$ with the constants and mixed norm of the sparse-grid theorem. Then ${({\sum }_{\alpha \in \mathrm{\Lambda }}\mathrm{\mid }{Q}_{{\mathrm{\Lambda }}_n}(u{\psi }_{\alpha })-⟨u,{\psi }_{\alpha }⟩{\mathrm{\mid }}^2)}^{1\mathrm{/}2}\le {\left({\sum }_{\alpha \in \mathrm{\Lambda }}{E}_{\alpha ,n}^2\right)}^{1\mathrm{/}2}\mathrm{.}$ The error-split theorem consequently bounds the L2 coefficient contribution by the right-hand side; this claim is restricted to the stated cube law and mixed regularity.