Strictly positive definite kernels on subsets of the complex plane

Let (z, w) ∈ ℂ × ℂ (zw) be a positive definite kernel and B a subset of ℂ. In this paper, we seek conditions in order that the restriction (z, w) ∈ B × B(zw) be strictly positive definite. Since this problem has been solved recently in the cases in which B is either ℂ or the unit circle in ℂ, our purpose here is twofold: to present some results we obtained when attempting to solve the problem for the above and other choices of B and to acquaint the audience with some other questions that remain. For two different classes of subsets, we completely characterize the strict positive definiteness of the kernel. We include a complete discussion of the case in which B is the unit circle of ℂ, making a comparison with the classical problem of strict positive definiteness on the real circle.