VARIOUS INEQUALITIES IN REPRODUCING KERNEL HILBERT SPACES

Abstract

In this paper, we examine various reproducing kernel Hilbert spaces $\mathcal{H}_{K_1}$ and $\mathcal{H}_{K_2}$ such that the inequality \[\det \left[\langle F_i G_i, F_j G_j \rangle_{\mathcal{H}_{K_1 K_2}} \right]_{i,j=1}^m \le C \det \left[\langle F_i,F_j \rangle_{\mathcal{H}_{K_1}} \langle G_i,G_j \rangle_{\mathcal{H}_{K_2}} \right]_{i,j=1}^m\] holds for all $F_j \in \mathcal{H}_{K_1}$, $G_j \in \mathcal{H}_{K_2}$, where $m$ is a positive integer, $C$ is a constant which is independent on $F_j$ and $G_j$ for all $j=1,2,...,m,$ and $\mathcal{H}_{K_1 K_2}$ is the Hilbert space admitting the reproducing kernel $K_1 K_2$.