We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study the relation between continuous and discrete frames in a random sampling regime, where discrete frames can be seen as Monte Carlo estimates of the continuous ones. Pairing spectral regularization with learning theory, we show that a sample frame tends to its population counterpart, and derive explicit finite-sample rates on spaces of Sobolev and Besov regularity. Our results prove the stability of frames constructed on empirical data, in the sense that all stochastic discretizations have the same underlying limit regardless of the set of initial training samples.

我们介绍了在一般域上的多尺度紧框架的构造。帧元素是通过对与再现内核相关的积分运算符进行频谱过滤而获得的。我们的构造在连续和离散非欧几里得结构（如黎曼流形和加权图）上扩展了经典小波以及广义小波。此外，它允许研究随机抽样方案中连续帧和离散帧之间的关系，其中离散帧可以看作是连续帧的蒙特卡洛估计。结合光谱正则化和学习理论，我们表明样本框架趋向于其人口对应物，并在Sobolev和Besov正则性的空间上得出显式的有限样本率。我们的结果证明了基于经验数据构建的框架的稳定性，