Exponential-type Bernstein-Kantorovich polynomials in q-analogue with graphical analysis

Abstract

Several authors have defined and studied various generalizations of the well-known Bernstein type operators. For functions that may not be continuous (according to Morigi and Neamtu), the traditional operators were replaced with their integral extensions defined in the Kantorovich and/ or Durrmeyer sense. In 2019, Aral et al. reconstructed the Bernstein-Kantorovich exponential-type polynomials and obtained some approximation results. In this work, we develop the q-analogue of the Bernstein-Kantorovich exponential-type operators. These newly defined operators provide an approximation process within exponentially weighted spaces, like Hp,& micro;[0, 1]. By employing the modulus of continuity alongside K-functionals, we obtain various precise quantitative estimates and further deduce a quantitative version of the Voronovskaja-type theorem. Finally, for graphical representation, we use MATLAB (R2025a).