In this work, we study the Hermite interpolation on $n$-dimensional non-equally spaced, rectilinear grids over a field $\Bbbk $ 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; we derive the ideal of the interpolation and express the interpolation remainder using only polynomial divisions, in the case of interpolating a polynomial function. Moreover, we prove the continuity of Hermite polynomials defined on adjacent $n$-dimensional grids, thus establishing spline behavior. 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$\Bbbk$上，针对$n$维非等距直线网格的埃尔米特插值问题，已知网格各点的函数值及直至最高阶的偏导数值。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解，该解仅使用单一求和形式，与现有$n$维经典埃尔米特插值的唯一替代闭式解[1]相比，代数形式更为简洁。我们给出了积分形式的插值余项；推导了插值理想，并在插值对象为多项式函数的情形下，仅通过多项式除法表达了插值余项。此外，我们证明了定义在相邻$n$维网格上的埃尔米特多项式的连续性，从而建立了样条行为。最后，我们通过数值算例展示了所提插值方法的适用性与高精度，涵盖少量点以及三维网格上数百点（采用类样条插值）的简单情形，其结果优于当前最先进的样条插值方法。