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We extend the univariate Newton interpolation algorithm to arbitrary spatialdimensions and for any choice of downward-closed polynomial space, whilepreserving its quadratic runtime and linear storage cost. The generalisationsupports any choice of the provided notion of non-tensorial unisolventinterpolation nodes, whose number coincides with the dimension of thechosen-downward closed space. Specifically, we prove that by selectingLeja-ordered Chebyshev-Lobatto or Leja nodes, the optimal geometricapproximation rates for a class of analytic functions -- termedBos--Levenberg--Trefethen functions -- are achieved and extend to thederivatives of the interpolants. In particular, choosing Euclidean degreeresults in downward-closed spaces whose dimension only grows sub-exponentiallywith spatial dimension, while delivering approximation rates close to, or evenmatching those of the tensorial maximum-degree case, mitigating the curse ofdimensionality. Several numerical experiments demonstrate the performance ofthe resulting multivariate Newton interpolation compared to state-of-the-artalternatives and validate our theoretical results.

Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions