Classical multivariate Hermite coordinate interpolation on n-dimensional grids

TL;DR

This work addresses multivariate Hermite interpolation on $n$-D non-equidistant grids by deriving a unique, dimension-free closed-form interpolant built from a non-fundamental basis and reverse-lexicographic derivative ordering. It provides a remainder formula via $n$-variate contour integrals and a Gröbner-basis description of the interpolation ideal, establishing deep algebraic structure for the method. A spline-oriented variant yields $C^0$ continuity (and higher when multiplicities are large) and offers practical, local computations that improve efficiency in high dimensions. Numerical experiments in 2D and 3D demonstrate superior accuracy compared to several state-of-the-art spline methods, highlighting potential applications in image resampling, geometric transforms, and multidimensional signal processing.

Abstract

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.

Figures (9)

• Figure 1: (a) The interpolated function and Hermite polynomial of the Example \ref{['hermerrr']}, along with the support points (b) Semi-logarithmic plot of the remainder of interpolation.
• Figure 2: Visualization of an example of spline local support points round a random point (green circle) in 2D, with $\lvert B_1 \rvert=\lvert B_2 \rvert=3$.
• Figure 3: Visualization of an example of spline local support points round a random point (green circle) in 3D, with $\lvert B_1 \rvert=\lvert B_2 \rvert=\lvert B_3 \rvert=3$.
• Figure 4: The two Hermite interpolants of example \ref{['examplesplinecont']} with the support points shown as red circles. The continuity of the Hermite spline along the points with $x_1=1$ is visually confirmed.
• Figure 5: The continuity of the interpolants of $f(x)$, with multiplicity functions $\nu(a) = \{1,2,3\}$, for every point $a$ on the grid $A$, and its derivatives up to order 3 of Example \ref{['exxxx']}. The different colors indicate the different spline interpolants for each interval. Dotted, dashed and continuous line indicates the interpolant created with $v(a)=1$, $v(a)=2$ and $v(a)=3$, respectively. (a) the $C^0$ continuity of the spline interpolants $f(x)$ (b) $C^1$ continuity of the spline interpolants of $f(x)$ (c) the $C^2$ continuity of the spline interpolants of $f(x)$ (d) the $C^3$ continuity of the spline interpolants of $f(x)$.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 2.1
• proof
• Theorem 2.2
• proof