In this paper, we consider the reproducing property in Reproducing Kernel Hilbert Spaces (RKHS). We establish a reproducing property for the closure of the class of combinations of composition operators under minimal conditions. This allows to revisit the sufficient conditions for the reproducing property to hold for the derivative operator, as well as for the existence of the mean embedding function. These results provide a framework of application of the representer theorem for regularized learning algorithms that involve data for function values, gradients, or any other operator from the considered class.

翻译：本文研究再生核希尔伯特空间（RKHS）中的再生性质。我们在最简条件下建立了复合算子线性组合集合闭包的再生性质。基于此，我们重新审视了微分算子满足再生性质的充分条件，以及均值嵌入函数存在的条件。这些结果为涉及函数值、梯度或该算子类中任意其他算子数据的正则化学习算法提供了表示定理的应用框架。