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 (Faber, 1908) 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 (Triebel, 2012) 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.

本文致力于建立一个高阶Faber样条基础，以比Lipschitz具有更高规则性的函数的样本离散化。除了支撑的紧凑性外，本文构建的基础与分段线性经典Faber-Schauder基础（Faber，1908年）具有相似的属性。尽管新的基本函数在实线上受支持，但它们的定位很好（呈指数衰减），主要部分集中在一个线段上。通过将经典的Faber基础扩展到更高的阶数，此构造可以完全解决Triebel专着（Triebel，2012年）中的问题3.13。粗略地讲，获得高阶Faber样条基的关键思想是将Taylor的余数公式应用于Chui-Wang对偶小波。第一步，我们明确确定这些双小波，它们可能具有独立的意义。利用这一新的基础，我们提供了Besov和Triebel–Lizorkin空间的采样特征，并克服了来自经典分段线性Faber–Schauder系统的平滑度限制。这个基础是无条件的，并且系数函数是根据类似于Faber–Schauder情况的离散函数值计算得出的。