Recently, samplets have been introduced as localized discrete signed measures which are tailored to an underlying data set. Samplets exhibit vanishing moments, i.e., their measure integrals vanish for all polynomials up to a certain degree, which allows for feature detection and data compression. In the present article, we extend the different construction steps of samplets to functionals in Banach spaces more general than point evaluations. To obtain stable representations, we assume that these functionals form frames with square-summable coefficients or even Riesz bases with square-summable coefficients. In either case, the corresponding analysis operator is injective and we obtain samplet bases with the desired properties by means of constructing an isometry of the analysis operator's image. Making the assumption that the dual of the Banach space under consideration is imbedded into the space of compactly supported distributions, the multilevel hierarchy for the generalized samplet construction is obtained by spectral clustering of a similarity graph for the functionals' supports. Based on this multilevel hierarchy, generalized samplets exhibit vanishing moments with respect to a given set of primitives within the Banach space. We derive an abstract localization result for the generalized samplet coefficients with respect to the samplets' support sizes and the approximability of the Banach space elements by the chosen primitives. Finally, we present three examples showcasing the generalized samplet framework.

翻译：近年来，样本小波被提出作为针对底层数据集定制的局部化离散符号测度。样本小波具有消失矩特性，即其测度积分对所有不超过特定次数的多项式均为零，这一特性可用于特征检测与数据压缩。本文中，我们将样本小波的不同构造步骤推广至比点赋值更一般的Banach空间泛函。为获得稳定表示，我们假设这些泛函构成具有平方可和系数的框架，甚至是具有平方可和系数的Riesz基。在两种情形下，相应的分析算子均为单射，通过构造分析算子像空间的等距映射，我们获得了具备所需性质的样本小波基。假设所考虑Banach空间的对偶空间嵌入至紧支集分布空间，则广义样本小波构造的多层层次结构可通过泛函支撑集的相似图谱聚类获得。基于该多层层次结构，广义样本小波对Banach空间内给定的一组基元具有消失矩特性。我们推导了广义样本小波系数关于样本小波支撑集尺寸及Banach空间元素被选定基元逼近能力的抽象局部化结果。最后，我们通过三个示例展示广义样本小波框架的应用。