Inequalities for Positive Definite Functions

Positive definite kernels and functions are considered. The key tool in the paper is the well-known main inequality for such kernels, namely, the Cauchy–Bunyakovskii inequality for the special inner product generated by a given positive definite kernel. It is shown that Ingham's inequality (and, in particular, Hilbert's inequality) is, essentially, the main inequality for the positive definite function $\sin(\pi x)/x$ on $\mathbb{R}$ and for a system of integer points. Using the main inequality, we prove new inequalities of Krein–Gorin type and Ingham's inequality.