We find probability error bounds for approximations of functions in a separable reproducing kernel Hilbert space with reproducing kernel on a base space , firstly in terms of finite linear combinations of functions of type and then in terms of the projection on , for random sequences of points in . Given a probability measure , letting be the measure defined by , , our approach is based on the nonexpansive operator where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by , that is the operator range of . Our main result establishes bounds, in terms of the operator , on the probability that the Hilbert space distance between an arbitrary function in and linear combinations of functions of type , for sampled independently from , falls below a given threshold. For sequences of points constituting a so-called uniqueness set, the orthogonal projections to converge in the strong operator topology to the identity operator. We prove that, under the assumption that is dense in , any sequence of points sampled independently from yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space are presented as well.

中文翻译：

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再现核希尔伯特空间中函数逼近的概率误差界

我们找到的函数逼近概率误差范围以可分离再生核Hilbert空间与再生核上的基空间，首先，在的类型的函数的有限线性组合的术语，然后在投影方面上，对于点的随机序列在。给定一个概率测度，又让被测量的定义， ，我们的方法是基于非扩展运算符在Bochner的意义上积分存在的地方。然后，使用该运算符定义一个新的再现内核Hilbert空间，用表示，该空间是的运算符范围。我们的主要结果建立边界，在运营商而言，对一个任意函数之间的希尔伯特空间距离的概率在和类型的函数的线性组合，用于从独立地采样，低于给定阈值时。对于构成所谓的唯一性集的点序列，正交投影以强算子拓扑收敛到身份算子。我们假设如果密集，则任何独立于点进行采样的点序列都将产生具有1概率的唯一性集。此结果在较弱的范数（例如统一或范数）上的先前误差范围有所改善，后者仅产生概率收敛，而几乎没有收敛。还给出了两个例子，显示了该结果适用于紧凑区间上的均匀分布以及哈代空间的适用性。