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.

翻译：本文研究了在特征为零的域k上，基于n维非等距直线网格，给定每个网格点上的函数值及直至最高阶的偏导数值的埃尔米特插值问题。首先，我们证明了插值多项式的唯一性，并进一步获得了一个紧凑的闭式表达式，该表达式仅使用单重求和，与维度无关。所得闭式公式的算术复杂度优于n维经典埃尔米特插值的唯一替代闭式形式[1]。此外，我们给出了插值的余项。最后，通过数值算例展示所提插值相较于其他插值方法的适用性和高精度。