Base-$b$ Haar contrasts and multiresolution ANOVA components

Let $b\ge 2$ and $d\ge 1$ be integers. For integers $k\ge 0$ and $0\le t<b^k$, put $I_{k,t}=[t{b}^{-k},(t+1)b^{-k})$. For an integer $1\le r<b$, define the normalized base-$b$ Haar contrast $h_{k,t,r}(x)=\sqrt{\frac{b^{k+1}}{r(r+1)}}({\sum }_{c=0}^{r-1}{\mathbf{1}}_{I_{k+1,bt+c}}(x)-r{\mathbf{1}}_{I_{k+1,bt+r}}(x))\mathrm{.}$

For a nonempty subset $u\subseteq \{1,\dots ,d\}$, let $\kappa =(k_j{)}_{j\in u}$, $\tau =(t_j{)}_{j\in u}$, and $\rho =(r_j{)}_{j\in u}$ have integer entries satisfying $k_j\ge 0$, $0\le t_j<b^{k_j}$, and $1\le r_j<b$. Set ${\psi }_{u,\kappa ,\tau ,\rho }(x)={\prod }_{j\in u}{h}_{k_j,t_j,r_j}(x_j)\mathrm{.}$ For a real Borel function $f\in L^2([0,1{)}^d)$, define $I(f)={\int }_{[0,1{)}^d}f(x)dx,{\nu }_{u,\kappa }(f)={\sum }_{\tau ,\rho }({\int }_{[0,1{)}^d}f(x){\psi }_{u,\kappa ,\tau ,\rho }(x)dx){\psi }_{u,\kappa ,\tau ,\rho },{\sigma }_{u,\kappa }^2(f)={\int }_{[0,1{)}^d}{\nu }_{u,\kappa }(f{)}^{2}dx\mathrm{.}$ The sums defining each ${\nu }_{u,\kappa }$ are finite. The complete orthogonal decomposition across all $u,\kappa$ is established separately.