Various Inequalities in Reproducing Kernel Hilbert Spaces

Abstract

In this paper, we examine various reproducing kernel Hilbert spaces H K1 and H K2 such that the inequalityholds for all F j ∈ H K1 , G j ∈ H K2 , 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 H K1K2 is the Hilbert space admitting the reproducing kernel K 1 K 2 .

Inequalities for Gram Matrices and Their Applications to Reproducing Kernel Hilbert Spaces

“…Applying the above theorem to the tensor product Hilbert space of RKHSs (Theorem 3.1), we can immediately obtain all the Gram determinant inequalities proved in [3]. Furthermore, we also obtain the equality conditions for these inequalities.…”

“…Recently, N. D. V. Nhan and D. T. Duc [3] have found interesting Gram determinant inequalities of the following form: For every F i ∈ H K 1 and G j ∈ H K 2 (i, j = 1, . .…”