This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber-Schauder basis except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebel's monograph (see References [47]) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylor's remainder formula to the dual Chui-Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel-Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber-Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber-Schauder situation.

中文翻译：

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用于函数采样离散化的高阶 Faber 样条基

本文致力于为具有比 Lipschitz 更高正则性的函数的采样离散化构建高阶 Faber 样条基的问题。本文构建的基与分段线性经典 Faber-Schauder 基具有相似的性质，除了支持的紧凑性。尽管在实线上支持新的基函数，但它们被很好地定位（指数衰减）并且主要部分集中在一个段上。通过将经典 Faber 基扩展到更高阶，该构造给出了 Triebel 专着（参见参考文献 [47]）中问题 3.13 的完整答案。粗略地说，获得高阶 Faber 样条基的关键思想是将泰勒余数公式应用于对偶 Chui-Wang 小波。作为第一步，我们明确地确定这些可能具有独立兴趣的双小波。使用这个新基础，我们为 Besov 和 Triebel-Lizorkin 空间提供了采样特征，并克服了来自经典分段线性 Faber-Schauder 系统的平滑度限制。该基是无条件的，系数泛函是根据与 Faber-Schauder 情况类似的离散函数值计算的。