On Kurre-Verma positive linear operators: convergence and asymptotics

Abstract

We introduce the Kurre--Verma (KV) positive linear operators on C[0,1], constructed via tempered binomial weights coupled with Beta-distribution integral kernels. This four-parameter structure enables precise variance control and boundary adaptivity, addressing limitations of classical schemes. We establish positivity, linearity, and normalization (preservation of constants) of KV operators, together with asymptotic reproduction of first and second moments. Uniform convergence is proved using Korovkin's theorem, with quantitative rates obtained using Peetre's K functional in connection with the modulus of continuity. A Voronovskaja-type asymptotic formula and a saturation theorem characterize the approximation order sharply. Numerical experiments illustrate the improved performance of KV operators compared with classical Jain--Kantorovich operators. These results highlight both the theoretical significance and the practical potential of the KV framework in approximation theory.