This paper begins by reviewing numerous theoretical advancements in the field of multivariate splines, primarily contributed by Professor Larry L. Schumaker. These foundational results have paved the way for a wide range of applications and computational techniques. The paper then proceeds to highlight various practical applications of multivariate splines. These include scattered data fitting and interpolation, the construction of smooth curves and surfaces, and the numerical solutions of various partial differential equations, encompassing both linear and nonlinear PDEs. Beyond these conventional and well-established uses, the paper introduces a novel application of multivariate splines in function value denoising. This innovative approach facilitates the creation of LKB splines, which are instrumental in approximating high-dimensional functions and effectively circumventing the curse of dimensionality.

翻译：本文首先回顾了多元样条领域的众多理论进展，这些进展主要归功于Larry L. Schumaker教授的贡献。这些基础性成果为广泛的应用和计算技术铺平了道路。随后，本文着重介绍了多元样条的各种实际应用，包括散乱数据拟合与插值、光滑曲线与曲面的构造，以及包括线性和非线性偏微分方程在内的多种偏微分方程的数值求解。除了这些传统且成熟的应用之外，本文还介绍了一种多元样条在函数值去噪中的新应用。这种创新方法有助于构建LKB样条，该样条在逼近高维函数方面发挥作用，并有效规避了维数灾难问题。