A Computationally-Efficient Closed-Form $C^{s,1}$-Extension Formula

We identify explicit closed-form and variational formulae for interpolating the exact values and derivatives through order $s$ of a $C^{s,1}$ function $f:{\mathbb{R}}^d\to \mathbb{R}$ at $N$ distinct points in $[0,1{]}^d$. The reconstruction satisfies bounds on its global $C^{s,1}$ seminorm and its Lipschitz constant on $[0,1{]}^d$ that are independent of the sample size $N$, the latter being the sharp Whitney condition. For $s\ge 2$, the reconstruction is real analytic away from the data points and definable in the o-minimal structure ${\mathbb{R}}_{\exp }$. Our nonlinear extension formula coincides with the formula of McShane (1934) for $s=0$ and with the formulae of Le Gruyer and Phan (2015) and Azagra, Le Gruyer, and Mudarra (2018) for $s=1$, and is new for $s\ge 2$. For fixed $d$ and $s\ge 2$, our closed-form formula is computable away from the data points by a circuit using only elementary unary and binary real operations, with $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ depth. This yields $\mathcal{O}(N)$ storage and $\mathcal{O}(\log N)$ parallel evaluation time. From the supplied coefficients and weights, the circuit can be compiled with $\mathcal{O}(N)$ one-time work in the exact-real word-RAM model. Thus, in this supplied-data setting, our nonlinear construction improves by a logarithmic factor on the $\mathcal{O}(N\log N)$ initialization bound of Fefferman and Klartag (2009), while retaining linear storage and attaining logarithmic query time through parallel evaluation.