Probability Error Bounds for Approximation of Functions in Reproducing Kernel Hilbert Spaces

[摘要] 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.

[学科分类] 护理学