The paper contains three main sections. In Sect. 2, the authors talk about various spaces of distributions, and eventually arrive at a Hilbert space \(X\) which is central to their discussion. They prove that this space is contained (only as a set) in \(C(\mathbb{R}^n)\) under appropriate hypothesis. In Sect. 3, they prove that this same space is actually embedded in \(C (\mathbb{R}^n) \), whenever it is a subspace of this same space. What they mean by this is that the natural linear mapping from \(X\) to \(C(\mathbb{R}^n)\) defined by the process of containment is in fact a continuous mapping. They then derive the reproducing kernel for \(X\).
Section 4 explains the connection with the idea of conditional positive definiteness. It also briefly discusses the interpolant which arises naturally in \(X\), and shows how to estimate the pointwise error in interpolant. This paper concentrates on developing a wide variety of spaces for which a variational theory is available. For each of these spaces, it is shown that, there is a natural choice of basis functions. It is also shown how the theory leads to efficient ways of calculating the interpolant and to new error estimates.