In this work, we study the Hermite interpolation on n-dimensional non-equally 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, which is algebraically simpler than the only alternative closed form for the n-dimensional classical Hermite interpolation [1]. We provide the remainder of the interpolation in integral form; moreover, we derive the ideal of the interpolation and express the interpolation remainder using only polynomial divisions, in the case of interpolating a polynomial function. Finally, we perform illustrative numerical examples to showcase the applicability and high accuracy of the proposed interpolant, in the simple case of few points, as well as hundreds of points on 3D-grids using a spline-like interpolation, which compares favorably to state-of-the-art spline interpolation methods.

翻译：本文研究了在特征为零的域k上，针对n维非等距直线网格的Hermite插值问题，已知函数在网格各点的值以及直至最高阶的偏导数。首先，我们证明了插值多项式的唯一性，并进一步得到一个仅用单次求和表示的紧凑闭式解，该解与维度无关，在代数上比现有的n维经典Hermite插值的唯一替代闭式解[1]更为简洁。我们给出了积分形式的插值余项；此外，推导了插值理想，并在插值多项式函数的情形下，仅通过多项式除法表达了插值余项。最后，我们通过若干数值示例展示了所提插值方法的适用性和高精度，包括少量点以及使用样条类插值在3D网格上数百个点的简单情况，其结果与当前最先进的样条插值方法相比具有优势。