Classical multivariate Hermite coordinate interpolation in n-dimensional grid

In this work, we study the Hermite interpolation on n-dimensional non-equal spaced, rectilinear grids over a field k of characteristic zero, given the values of the function at each point of the grid and the partial derivatives up to a maximum degree. First, we prove the uniqueness of the interpolating polynomial, and we further obtain a compact closed form that uses a single summation, irrespective of the dimensionality. The arithmetic complexity of the derived closed formula compares favourably with the only alternative closed form for the n-dimensional classical Hermite interpolation [1]. In addition, we provide the remainder of the interpolation. Finally, we perform illustrative numerical examples to showcase the applicability and high accuracy of the proposed interpolant, compared to other interpolation methods.