Samplets are data adapted multiresolution analyses of localized discrete signed measures. They can be constructed on scattered data sites in arbitrary dimension and 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, when choosing polynomials as primitives, the resulting samplet basis converges in the infinite data limit to signed measures with broken polynomial densities. 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：Samplet是针对局部离散符号测度的数据自适应多分辨率分析方法。该方法可在任意维度的散乱数据点上构建，且能对任意指定基函数集合具有消失矩性质。我们考虑概率框架下的Samplet构造，并证明：当选择多项式作为基函数时，所得Samplet基在无限数据极限下收敛于具有分片多项式密度的符号测度。这些密度对应于数据点所在区域层次化剖分上的多小波。由此，我们获得了一种通用多小波的构造方法，该方法允许灵活指定超越张量积构造的消失矩性质。对于一致剖分，我们特别还原了具有尺度和剖分无关滤波器系数的经典多小波。理论结果由数值实验补充，这些实验展示了随机数据点和低差异数据点情况下的收敛性结论。