Samplets are data adapted multiresolution analyses of localized discrete signed measures. They can be constructed on scattered data sites in arbitrary dimension such that they exhibit vanishing moments with respect to any prescribed set of primitives. We consider the samplet construction in a probabilistic framework and show that, if choosing polynomials as primitives, the resulting samplet basis converges to signed measures with broken polynomial densities in the infinite data limit. These densities amount to multiwavelets with respect to a hierarchical partition of the region containing the data sites. As a byproduct, we therefore obtain a construction of general multiwavelets that allows for a flexible prescription of vanishing moments going beyond tensor product constructions. For congruent partitions we particularly recover classical multiwavelets with scale- and partition- independent filter coefficients. The theoretical findings are complemented by numerical experiments that illustrate the convergence results in case of random as well as low-discrepancy data sites.

翻译：样本限制（Samplets）是定义在离散符号测度上的局部化自适应多分辨率分析。它们可以在任意维度的散乱数据点上构造，从而对任意给定的原始函数集具有消失矩性质。我们在概率框架下研究样本限制的构造，并证明：若选择多项式作为原始函数，则当数据点数趋于无穷时，所得样本限制基将收敛于具有分段多项式密度的符号测度。这些密度对应于包含数据点的区域层次划分下的多小波。因此，作为副产品，我们得到了一种通用多小波构造方法，该方法允许灵活指定超越张量积构造的消失矩。对于全等划分，我们特别恢复了具有尺度无关和划分无关滤波器系数的经典多小波。理论发现辅以数值实验，展示了在随机和低差异数据点情况下的收敛结果。