Approximation Theorems and $\mathcal{I}$-Equi Statistical Convergence

In the present work, a novel type of convergence is introduced by incorporating the notions of $\mathcal{I}$-statistical convergence and equi-statistical convergence. These offer a more extensive framework in comparison to the classical concepts of $\mathcal{I}$-convergence and statistical convergence. Within this generalized setting, two fundamental approximation results are established and proven: one in the sense of Korovkin-type theorems and another in the sense of a Voronovskaya-type result. Moreover, the paper provides a concrete example that illustrates the practical significance of the new convergence concept. Finally, an estimate for the rate at which $\mathcal{I}$-equi-statistical convergence occurs is provided, thus quantifying the efficiency of approximation under this framework.