In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice is by no means mandatory and different choices, like, for example, the Lagrange basis, are possible and might offer additional features. In this article, we discuss different alternatives together with their canonical duals. We study a localized version of the Lagrange basis, localized orthogonal bases, such as the Newton basis, and multiresolution versions thereof, constructed by means of samplets. We argue that the choice of orthogonal bases is particularly useful as they lead to symmetric preconditioners. All bases under consideration are compared numerically to illustrate their feasibility for scattered data approximation. We provide benchmark experiments in two spatial dimensions and consider the reconstruction of an implicit surface as a relevant application from computer graphics.

翻译：在散乱数据逼近中，通常选取径向基函数的有限个平移张成的空间作为逼近空间，并采用平移基进行逼近表示。然而，这种自然选择并非强制要求，采用不同基函数（例如拉格朗日基）是可行的，且可能提供额外优势。本文探讨了多种替代方案及其规范对偶基。我们研究了拉格朗日基的局部化形式、局部正交基（如牛顿基）及其通过样本小波构造的多分辨率版本。我们认为正交基的选择尤其重要，因其可导出对称预条件子。通过数值比较所有讨论的基函数，阐明其在散乱数据逼近中的可行性。我们在二维空间中进行基准实验，并以计算机图形学中隐式曲面重建作为典型应用进行验证。