Korovkin-type approximation for non-positive operators
The classical Korovkin theorem traditionally relies on the positivity of the underlying sequence of operators. In 1968, D. E. Wulbert obtained a non-positive version by exploiting geometric properties of function spaces, namely the Choquet boundary and the unique extension property of extreme points of the dual unit ball for weakly separating subspaces. In this article we develop this geometric approach further and prove a Korovkin-type theorem for uniformly bounded sequences of operators on $C(X)$ and on $L^1[0,1]$, with the convergence of the operators on a test set being replaced by convergence to a limit operator. The main emphasis is on the underlying geometry: the weakly separating subspace, its Choquet boundary, and the associated unique extension property. As an application, we show that the Grünwald interpolation operators, which are non-positive, satisfy a Korovkin-type approximation theorem. We also extend this to an $L^1(\mathbb{R})$ setting and include numerical illustrations.