Which spaces can be embedded in Reproducing Kernel Hilbert Spaces?

Max Schölpple, Ingo Steinwart

Abstract:

Given a Banach space E consisting of functions, we ask whether there exists a reproducing kernel Hilbert
space H with bounded kernel such that E ⊂ H. More generally, we consider the question, whether for a given
Banach space consisting of functions F with E ⊂ F , there exists an intermediate reproducing kernel Hilbert
space E ⊂ H ⊂ F . We provide both sufficient and necessary conditions for this to hold. Moreover, we
show that for typical classes of function spaces described by smoothness there is a strong dependence on the
underlying dimension: the smoothness s required for the space E needs to grow proportional to the dimension
d in order to allow for an intermediate reproducing kernel Hilbert space H.