Anchored-box summation by parts for finite Hardy–Krause variation

Write the grid knots in coordinate $j$ as $0=t_{j,0}<\cdots <t_{j,m_j}=1$, and assign each point of $P$ to its unique half-open cell $C_k={\prod }_j[t_{j,k_j},t_{j,k_j+1})$. Define $c_{\mathrm{\ell }}=f(t_{1,{\mathrm{\ell }}_1},\dots ,t_{d,{\mathrm{\ell }}_d})$ for every vertex index with $0\le {\mathrm{\ell }}_j\le m_j$; cell tags use the lower indices $k_j<m_j$. Put $F(k)=N^{-1}\mathrm{\#}\{i:x_i\in {\prod }_j[0,t_{j,k_j})\}-{\prod }_j{t}_{j,k_j}$. Inclusion–exclusion in the $d$ coordinates gives $N^{-1}\mathrm{\#}\{i:x_i\in C_k\}-{\prod }_j(t_{j,k_j+1}-t_{j,k_j})={\mathrm{\Delta }}_1\cdots {\mathrm{\Delta }}_{d}F(k)$, with ${\mathrm{\Delta }}_{j}F(k)=F(k+e_j)-F(k)$. Also $F(k)=0$ if any $k_j=0$, while $F(m_1,\dots ,m_d)=0$ because all points lie in $[0,1{)}^d$ and the full cube has volume $1$. The grid contains every node coordinate, so the point's assigned cell has it as lower corner. Summing the cell identities weighted by $c_k$ gives $Q_P(f)-S_G(f)={\sum }_k{c}_k{\mathrm{\Delta }}_1\cdots {\mathrm{\Delta }}_{d}F(k)$.

For one coordinate, the finite summation-by-parts identity is ${\sum }_{k=0}^{m-1}{a}_k(b_{k+1}-b_k)=a_m{b}_m-a_0{b}_0-{\sum }_{k=1}^m{b}_k(a_k-a_{k-1})\mathrm{.}$ In each application here, the sequence $b$ has $b_0=0$: initially it is $F$ with that coordinate index zero, and at each successive application it is a finite difference in other coordinates of such values of $F$. Step 1.1 gives $F=0$ whenever any coordinate index is zero, so these lower-boundary terms vanish throughout the iteration and the shortened identity with no $a_0{b}_0$ term is valid. Applying it successively in all $d$ coordinates gives a sum over subsets $u\subseteq \{1,\dots ,d\}$: the empty subset contributes $c_{m_1,\dots ,m_d}F(m_1,\dots ,m_d)=0$; for each nonempty $u$, the terms are $(-1{)}^{\mathrm{\mid }u\mathrm{\mid }}F(k_u,m_{-u}){\mathrm{\Delta }}_{u}c(k_u,m_{-u})$, summed over $1\le k_j\le m_j$ for $j\in u$. Here ${\mathrm{\Delta }}_{u}c(k_u,m_{-u})$ is the alternating sum of the $2^{\mathrm{\mid }u\mathrm{\mid }}$ vertex values with coordinate index $k_j-1+{ϵ}_j$ for $j\in u$ and index $m_j$ for $j\mathrm{\notin }u$, weighted by $(-1{)}^{\mathrm{\mid }u\mathrm{\mid }-\mathrm{\mid }ϵ\mathrm{\mid }}$. The identity follows by induction on the number of coordinates: the one-coordinate formula is the base case, and applying it in the next finite sum adds that coordinate to the subsets in the difference terms while giving the stated upper-boundary term. Each $F(k_u,m_{-u})$ is the discrepancy of the anchored box ${\prod }_{j\in u}[0,t_{j,k_j})\times {\prod }_{j\mathrm{\notin }u}[0,1)$, so its absolute value is at most $D_N^{\ast }(P)$ by [F1]. Each ${\mathrm{\Delta }}_{u}c$ is exactly the alternating increment on a cell of the face with complementary coordinates fixed at $1$; hence its absolute-value sum over that face grid is at most $V_V(f_u)$. Summing over nonempty $u$ and using [F2] gives $\mid Q_P(f)-S_G(f)\mid \le D_N^{\ast }(P)V_{HK,1}(f)$.