Positive partition of unity and interpolation error for periodic multilinear interpolation

The supplier result in [F1] says each evaluation is a convex combination of the corner data. Applying this to $W-V\ge 0$ proves order preservation, and applying it to $V$ and $-V$ proves the sup-norm bound.

In one coordinate, let $x=x_i+\theta h$ with $0\le \theta \le 1$ and let $L_h\varphi (x)=(1-\theta )\varphi (x_i)+\theta \varphi (x_i+h)$. Taylor-expand both endpoints about $x$. By [F2], the linear terms cancel after weighting and $\mathrm{\mid }{L}_h\varphi (x)-\varphi (x)\mathrm{\mid }\le \frac{h^2}{2}\mathrm{\parallel }{\varphi }^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}((1-\theta ){\theta }^2+\theta (1-\theta {)}^2)=\frac{h^2}{2}\theta (1-\theta )\mathrm{\parallel }{\varphi }^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\le \frac{h^2}{8}\mathrm{\parallel }{\varphi }^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$ This calculation also holds on the periodic seam by taking a periodic lift of the cell.

Write the tensor interpolant as the product of the one-coordinate interpolation operators $I_j$. Each $I_j$ has sup-norm at most one by step 1.1. The identity $I_1\cdots I_d-I={\sum }_{j=1}^d{I}_1\cdots I_{j-1}(I_j-I)$ and the one-dimensional bound of step 1.2 give the claimed sum of errors. This proves the statement.