Smoothness bounds the high-frequency ANOVA components

If $u$ is the full coordinate set, then $f_u=f$ and ${\mathrm{\partial }}_u{f}_u={\mathrm{\partial }}_{u}f$. Otherwise, [F6] makes $f_u$ continuous: changing $x_u$ by less than $\delta$ changes the integrand uniformly in $x_{-u}$, and integration over a unit-volume cube preserves that bound. Repeated application of [F2] to the complementary coordinates, with [F4] identifying iterated integrals and [A1, F5] identifying the Riemann and Lebesgue integrals, gives ${\mathrm{\partial }}_u{f}_u(x_u)={\int }_{[0,1{]}^{d-\mathrm{\mid }u\mathrm{\mid }}}{\mathrm{\partial }}_{u}f(x_u,x_{-u})d{x}_{-u}$. The same uniform-continuity argument makes this derivative continuous. Since $\mathrm{\mid }{\mathrm{\partial }}_{u}f\mathrm{\mid }$ is continuous on the nonempty compact cube, [F7] gives a finite supremum; the displayed integral (or $f_u=f$ in the full-set case) shows $M_u\le {\sup }_{[0,1{]}^d}\mathrm{\mid }{\mathrm{\partial }}_{u}f\mathrm{\mid }$.

Fix $(\tau ,\rho )$ and let $c_{u,\kappa ,\tau ,\rho }=\int f_u(x_u){\prod }_{j\in u}{h}_{k_j,t_j,r_j}(x_j)d{x}_u$. The integrand is bounded and continuous on each cell of a finite rectangular partition, so it is Riemann integrable; by [A1, F5] this coefficient is also its Riemann integral. Each contrast has integral zero on its parent. Subtracting the left-endpoint value in one coordinate therefore leaves the integral unchanged. Applying [F3] successively in every coordinate expresses the resulting alternating finite difference as the iterated integral of ${\mathrm{\partial }}_u{f}_u$ over ${\prod }_{j\in u}[a_j,x_j]$; its absolute value is at most $M_u{\prod }_{j\in u}{b}^{-k_j}$. The outer integral of the absolute contrast product factors by [F4] as ${\prod }_{j\in u}\mathrm{\parallel }{h}_{k_j,t_j,r_j}{\mathrm{\parallel }}_1$.

From the explicit child values in [F1], for $1\le r<b$, $\mathrm{\parallel }{h}_{k,t,r}{\mathrm{\parallel }}_1=2\sqrt{\frac{r}{r+1}}{b}^{-(k+1)\mathrm{/}2}\le 2b^{-k\mathrm{/}2}\mathrm{.}$ Thus step 2.1 gives $\mathrm{\mid }{c}_{u,\kappa ,\tau ,\rho }\mathrm{\mid }\le 2^{\mathrm{\mid }u\mathrm{\mid }}{M}_u{b}^{-3\mathrm{\mid }\kappa \mathrm{\mid }\mathrm{/}2}$.

For fixed $u,\kappa$, the contrasts at distinct translations have disjoint supports, and at the same parent their child coefficient vectors are orthonormal by the dot-product calculation in [F1]. Their tensor products are therefore orthonormal. There are $b^{\mathrm{\mid }\kappa \mathrm{\mid }}(b-1{)}^{\mathrm{\mid }u\mathrm{\mid }}$ such products, so ${\sigma }_{u,\kappa }^2={\sum }_{\tau ,\rho }\mathrm{\mid }{c}_{u,\kappa ,\tau ,\rho }{\mathrm{\mid }}^2\le b^{\mathrm{\mid }\kappa \mathrm{\mid }}(b-1{)}^{\mathrm{\mid }u\mathrm{\mid }}{4}^{\mathrm{\mid }u\mathrm{\mid }}{M}_u^2{b}^{-3\mathrm{\mid }\kappa \mathrm{\mid }}=4^{\mathrm{\mid }u\mathrm{\mid }}(b-1{)}^{\mathrm{\mid }u\mathrm{\mid }}{M}_u^2{b}^{-2\mathrm{\mid }\kappa \mathrm{\mid }}$, as claimed.