Strictly positive definite kernels on subsets of the complex plane

In this paper we seek for inner product dependent strictly positive definite kernels on subsets of C. We present separated necessary and sufficient conditions in order that a positive definite kernel on C be strictly positive definite. One emphasis is on strictly positive definite kernels on the unit circle. Since positive definite kernels on the circle were already characterized in [1], the study in this case reduces to the determination of what kind of positive definite kernels are indeed strictly positive definite. For other subsets, we begin with a quite general positive definite kernel on the whole C and find conditions in order that it is strictly positive definite on the subset. For some classes of subsets, the results are final.