Probability error bounds for approximation of functions in reproducing kernel Hilbert spaces

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 Kxi and then in terms of the projection πxn on spanKxii=1n, for random sequences of points x=xii in X. Given a probability measure P, letting PK be the measure defined by dPKx=Kx,xdPx, x∈X, our approach is based on the nonexpansive operator L2X;PK∋λ→LP,Kλ≔∫XλxKxdPx∈H, where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by HP, that is the operator range of LP,K. Our main result establishes bounds, in terms of the operator LP,K, on the probability that the Hilbert space distance between an arbitrary function f in H and linear combinations of functions of type Kxi, for xii sampled independently from P, falls below a given threshold. For sequences of points xii=1∞ constituting a so-called uniqueness set, the orthogonal projections πxn to spanKxii=1n converge in the strong operator topology to the identity operator. We prove that, under the assumption that HP 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 Lp 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 H2D are presented as well.