Approximation by exponential-type polynomials

In this paper, a family of exponential-type polynomials is introduced and studied. Both the uniform and the L-p convergence are established in suitable function spaces. In the L-p-case, some estimates are also achieved using an exponentially weighted version of the p-norm. Further, a Voronovskaja type formula is proved, finding the exact order of pointwise approximation in case of continuous functions having second derivative at some points. Finally, quantitative estimates for the order of approximation are established in both the continuous and the L-p-cases in terms of the modulus of continuity and K-functionals of the involved functions. In the latter estimate, a crucial role is played by the so-called Hardy-Littlewood maximal function.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

Automated summary

This paper introduces and studies a family of exponential-type polynomials, and analyzes their uniform and L-p convergence in suitable function spaces. Estimates are achieved using an exponentially weighted version of the p-norm in the L-p-case. The paper also proves a Voronovskaja type formula to determine the exact order of pointwise approximation for continuous functions with second derivative at some points. Additionally, quantitative estimates for the order of approximation in both the continuous and L-p-cases are established based on the modulus of continuity and K-functionals, with a crucial role played by the Hardy-Littlewood maximal function.