This paper addresses the problem of regression to reconstruct functions, which are observed with superimposed errors at random locations. We address the problem in reproducing kernel Hilbert spaces. It is demonstrated that the estimator, which is often derived by employing Gaussian random fields, converges in the mean norm of the reproducing kernel Hilbert space to the conditional expectation and this implies local and uniform convergence of this function estimator. By preselecting the kernel, the problem does not suffer from the curse of dimensionality. The paper analyzes the statistical properties of the estimator. We derive convergence properties and provide a conservative rate of convergence for increasing sample sizes.