Absolute detail reconstruction and omitted-tail error bound

Choose $L$ large enough that $\mathrm{\Lambda }\subseteq R_L$. By [F1], [F3], and [F6], $Q_L^{\otimes d}(f)-Q_{\mathrm{\Lambda }}(f)={\sum }_{\mathrm{\ell }\in R_L\setminus \mathrm{\Lambda }}{\delta }_{\mathrm{\ell }}(f)$. The index set on the right is finite and is contained in ${\mathbb{N}}_0^d\setminus \mathrm{\Lambda }$.

By [F5] and [F2], the absolute value of the right side in step 1.1 is at most the omitted-detail tail $T_{\mathrm{\Lambda }}:={\sum }_{\mathrm{\ell }\mathrm{\notin }\mathrm{\Lambda }}\mathrm{\mid }{\delta }_{\mathrm{\ell }}(f)\mathrm{\mid }$ for every sufficiently large $L$. Put $x_L:=Q_L^{\otimes d}(f)-Q_{\mathrm{\Lambda }}(f)$ and $z:=I_d(f)-Q_{\mathrm{\Lambda }}(f)$; the convergence hypothesis in Given says $\mathrm{\mid }{x}_L-z\mathrm{\mid }=\mathrm{\mid }{Q}_L^{\otimes d}(f)-I_d(f)\mathrm{\mid }$ is eventually below every positive rational tolerance. If $\mathrm{\mid }z\mathrm{\mid }>T_{\mathrm{\Lambda }}$, [F10] supplies a positive rational $\epsilon$ with $2\epsilon <\mathrm{\mid }z\mathrm{\mid }-T_{\mathrm{\Lambda }}$. For sufficiently large $L$, both $\mathrm{\Lambda }\subseteq R_L$ and $\mathrm{\mid }{x}_L-z\mathrm{\mid }<\epsilon$ hold by finiteness of $\mathrm{\Lambda }$ and [F7]. The triangle inequality [F8] then gives $\mathrm{\mid }{x}_L\mathrm{\mid }\ge \mathrm{\mid }z\mathrm{\mid }-\mathrm{\mid }{x}_L-z\mathrm{\mid }>\mathrm{\mid }z\mathrm{\mid }-\epsilon >T_{\mathrm{\Lambda }}$, contradicting the preceding tail bound. Hence the displayed omitted-tail inequality holds.

For the dyadic composite-Simpson hierarchy, [F4] supplies the convergence hypothesis in Given. If the mixed-derivative hypotheses of [F9] hold, its common bound makes the finite subsums of ${\sum }_{\mathrm{\ell }\in {\mathbb{N}}_0^d}\mathrm{\mid }{\delta }_{\mathrm{\ell }}(f)\mathrm{\mid }$ bounded above. By the definition [F2], the full absolute-detail sum is therefore defined and finite, so step 2.1 applies and proves the stated particular error case.