Periodic tensor grids and multilinear interpolation

Let $d\ge 1$, let ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_d>0$, and let $N_1,\dots ,N_d\ge 1$ be integers. The periodic rectangular box is ${\mathbb{T}}_{\mathrm{\ell }}^d:={\mathbb{R}}^d\mathrm{/}({\mathrm{\ell }}_1\mathbb{Z}\times \cdots \times {\mathrm{\ell }}_d\mathbb{Z})\mathrm{.}$ Set $h_j:={\mathrm{\ell }}_j\mathrm{/}{N}_j$. Its uniform tensor grid has nodes $x_{\mathit{i}}=(i_1{h}_1,\dots ,i_d{h}_d)$, indexed by $0\le i_j<N_j$, with indices understood modulo $N_j$.

For node data $V_{\mathit{i}}$ and a lift $x\in {\mathbb{R}}^d$ of a point on the torus, define $q_j:=\lfloor x_j\mathrm{/}{h}_j\rfloor ,{\theta }_j:=x_j\mathrm{/}{h}_j-q_j\in [0,1),$ and for $\mathit{ϵ}\in \{0,1{\}}^d$ define $w_{\mathit{ϵ}}(x):={\prod }_{j=1}^d{\theta }_j^{{ϵ}_j}(1-{\theta }_j{)}^{1-{ϵ}_j}\mathrm{.}$ The tensor-product multilinear interpolant is ${\mathcal{I}}_{h}V(x):={\sum }_{\mathit{ϵ}\in \{0,1{\}}^d}{w}_{\mathit{ϵ}}(x)V_{([q_1+{ϵ}_1{]}_{N_1},\dots ,[q_d+{ϵ}_d{]}_{N_d})},$ where $[k{]}_N$ is the residue in $\{0,\dots ,N-1\}$. The formula is unchanged when a period ${\mathrm{\ell }}_j$ is added to $x_j$, so it defines an interpolant on ${\mathbb{T}}_{\mathrm{\ell }}^d$. If $N_j=1$, the two corner indices in that coordinate coincide; the formula still gives the same node value.

For a continuous periodic function $v$, ${\mathcal{I}}_{h}v$ means the interpolant of its grid samples. Put $h:={\max }_j{h}_j$.

The picture shows one lifted coordinate cell. The weights are $1-\theta$ at the left endpoint and $\theta$ at the right endpoint; periodic wrapping only changes the node indices.

The diagram fixes the coordinate and weight convention only. The multidimensional interpolant is the product of the coordinate weights in the displayed formula.