turkmath.org

We find probability error bounds for approximations of functions \(f\) in a separable reproducing kernel Hilbert space \(\mathcal{H}\) 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 \(\pi_x^n\)  on \(\text{span}\left\{ K_{x_i} \right\}_{i=1}^{n}\),  or random sequences of points \(x = (x_i)_i\) in \(X\). Given a probability measure \(P\), letting \(P_K\) be the measure defined by \(\text{d}P_K(x) = K(x, x)\text{d}P(x), \ x\in X\), our approach is based on the noneexpansive operator
                 
                        \[L^2(X; P_K) \ni \lambda \mapsto L_{P,K} \lambda :=   \int_X \lambda(x)\, K_x\, \text{d}P(x) \in \mathcal{H},\]

where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by \(\mathcal{H}_P\), that is the operator range of \(L_{P,K}\). Our main result establishes bounds, in terms of the operator \(L_{P,K}\),  on the probability  that the Hilbert space distance between an arbitrary function \(f\) in \(\mathcal{H}\) and linear combinations of functions of type \(K_{x_i}\), for \((x_i)_i\), sampled independently from \(P\), falls below a given threshold. For sequences of points \((x_i)_i^\infty\) constituting a so-called uniqueness set, the orthogonal projections \(\pi_x^n\) to \(\text{span}\left\{ K_{x_i} \right\}_{i=1}^{n}\) converge in the strong operator topology to the identity operator. We prove that, under the assumption that \(\mathcal{H}_P\) is dense in \(\mathcal{H}\),  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 \(H^2(\mathbb{D})\) are presented as well.