Pasting Reproducing Kernel Hilbert Spaces

Abstract

The aim of this article is to find the necessary and sufficient condition for the mapping

$$\displaystyle{H_{K}(E) \ni f\mapsto (\,f\vert E_{1},f\vert E_{2}) \in H_{K\vert E_{1}\times E_{2}}(E_{1}) \oplus H_{K\vert E_{2}\times E_{2}}(E_{2})}$$

to be isomorphic, where K is a positive definite function on E = E ₁ + E ₂. As an application, the Binet-Cauchy equality and its variant are considered.