Non-asymptotic confidence regions on RKHS. The Paley-Wiener and standard Sobolev space cases

We consider the problem of constructing a global, probabilistic, and non-asymptotic confidence region for an unknown function observed on a random design. The unknown function is assumed to lie in a reproducing kernel Hilbert space (RKHS). We show that this construction can be reduced to accurately estimating the RKHS norm of the unknown function. Our analysis primarily focuses both on the Paley-Wiener and on the standard Sobolev space settings.