The Fast Newton Transform: Interpolation in Downward Closed Polynomial Spaces

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 with respect to $\ell^p$-sets $A_{m,n,p}$, in non-tensorial Leja ordered Chebyshev-Lobatto or Leja grids. Especially, the $\ell^2$-Euclidean case $A_{m,n,2}$ turns out to be the pivotal choice to mitigate the curse of dimensionality, leading to a ratio $|A_{m,n,2}| / |A_{m,n,\infty}|$ that decays exponentially with spatial dimension $m$, while reaching close to or the same approximation power as the tensorial $\ell^\infty$-case. Expanding non-periodic functions, the FNT complements the approximation capabilities of the Fast Fourier Transform (FFT), whereas the choice of $\ell^p$-sets renders the FNT time complexity to be less than the FFT time complexity in a wide range of $n$, that exponentially increases with $m$. Maintaining this advantage true for the differentials, the FNT sets a new standard in $m$-variate interpolation and approximation practice.