The Embedding Theory of Native Spaces

In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This space can be completed and become a Hilbert space, called native space, which is of great importance in the last decade. The native space then contains some abstract elements which are not linear combinations of radial basis functions. The meaning of these abstract elements is not fully known. This paper presents some interpretations for the these elements. The native spaces are embedded into some well-known spaces. For example, the Sobolev-space is shown to be a native space. Since many differential equations have solutions in the Sobolev-space, we can therefore approximate the solutions by linear combinations of radial basis functions. Moreover, the famous question of the embedding of the native space into L²(Ω) is also solved by the author.     

On the Hausdorff-limitability of {zn}

In this paper we are concerned with the Hausdorff-limitability of the sequence {zⁿ}. Our aim is to solve a problem, which was posed by Agnew [1]. The H_x-transform of {zⁿ}: $$\sigma _n (z) = \int\limits_0^1 {\{ 1 + t(z - 1)\} ^n d\chi (t)}$$ (where: χ∈bv(0,1), χ(0)=χ(0+)=0, χ(1)=1) converges to the value zero for \(z \in r_r = \{ \varsigma :|\varsigma - (1 - \tfrac{1}{r})| = \tfrac{1}{r},\varsigma = |1\} (o< r \leqslant 1)\) if and only if χ satisfies the conditions: χ(t)=1 for t∈[r, 1] and χ is continuous at t=r.