Approximation by exponential-type polynomials (Q6181163)

The authors introduce and examine a family of exponential-type polynomials. They establish the uniform convergence and the \(L^p\) convergence in suitable function spaces. Certain estimations in the \(L^p\)-case are also obtained by means of an exponentially weighted \(p\)-norm version. Furthermore, a formula of the Voronovskaja type is established by the authors, which determines the exact order of pointwise approximation when dealing with continuous functions that have a second derivative at certain points. Additionally, they use the modulus of continuity and \(K\)-functionals of the involved functions to establish quantitative estimates for the order of approximation in both the continuous and the \(L^p\)-case. In the end, the Hardy-Littlewood maximum function is seen as having a pivotal role.