The Fast Newton Transform (FNT) addresses the computational bottleneck that arises in solving high-dimensional problems such as 6d Boltzmann, Fokker-Planck, or Vlaslov equations, multi-body Hamiltonian systems, and the inference of governing equations in complex self-organizing systems. Specifically, the challenge lies in numerically computing function expansions and their derivatives fast, while achieving high approximation power. The FNT is a Newton interpolation algorithm with runtime complexity , where N is the dimension of the downward closed polynomial space, its degree and the spatial dimension. We select subgrids from tensorial Leja-ordered Chebyshev-Lobatto grids based on downward-closed sets. This significantly reduces the number of coefficients, , while achieving optimal geometric approximation rates for a class of analytic functions known as Bos–Levenberg–Trefethen functions. Specifically, we investigate -multi-index sets, where the Euclidean degree () turns out to be the pivotal choice for mitigating the curse of dimensionality. Furthermore, the differentiation matrices in Newton basis are sparse, enabling the implementation of fast pseudo-spectral methods on flat spaces, polygonal domains, and regular manifolds. Numerical experiments validate the algorithm's superior runtime performance over state-of-the-art approaches.