The Fast Newton Transform: Interpolation in Downward Closed Polynomial Spaces

Abstract: We present the Fast Newton Transform (FNT), an algorithm for performing $m$-variate Newton interpolation in downward closed polynomial spaces with time complexity $\mathcal{O}(|A|m\overline{n})$. Here, $A$ is a downward closed set of cardinality $|A|$ equal to the dimension of the associated downward closed polynomial space $\Pi_A$, where $\overline{n}$ denotes the mean of the maximum polynomial degrees across the spatial dimensions $m$. For functions being analytic in an open Bernstein poly-ellipse, geometric approximation rates apply when interpolating in non-tensorial Leja-ordered Chebyshev-Lobatto or Leja grids. To mitigate the curse of dimensionality, we utilize $\ell^p$-sets, with the $l^2$-Euclidean case turning out to be the pivotal choice, leading to $|A|/(n+1)^m \in \mathcal{O}(e^{-m})$. Expanding non-periodic functions, the FNT complements the approximation capabilities of the Fast Fourier Transform (FFT). Choosing $\ell^2$-sets for $A$ renders the FNT time complexity to be less than the FFT time complexity $\mathcal{O}((n+1)^m m \log(n))$ in a range of $n$, behaving as $\mathcal{O}(e^m)$. Maintaining this advantage true for the differentials, the FNT sets a new standard in $m$-variate interpolation and approximation practice.