Strictly positive definite kernels on subsets of the complex plane

Let (z, w) × (zw) be a positive definite kernel and B a subset of . 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 or the unit circle in , 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 , making a comparison with the classical problem of strict positive definiteness on the real circle.