Reproducing Kernel Hilbert Spaces Approximation Bounds
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We find probability error bounds for approximations of functions f in a separable reproducing kernel Hilbert space H with reproducing kernel K on a base space X, firstly in terms of finite linear combinations of functions of type K_x_i and then in terms of the projection π^n_x on Span{K_x_i}^n_i=1, for random sequences of points x=(x_i)_i in the base space X. Previous results demonstrate that, for sequences of points (x_i)_i=1^∞ constituting a so-called uniqueness set, the orthogonal projections π^n_x to Span{K_x_i}^n_i=1 converge in the strong operator topology to the identity operator. The main result shows that, for a given probability measure P, letting P_K be the measure defined by d P_K(x)=K(x,x)d P(x), x∈ X, and H_P denote the reproducing kernel Hilbert space that is the operator range of the nonexpansive operator L^2(X;P_K)∋λ L_P,Kλ:=∫_X λ(x)K_xd P(x)∈H, where the integral exists in the Bochner sense, under the assumption that H_P is dense in H any sequence of points sampled independently from P yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or L^p 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 H^2(D) are presented as well.
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