On the uniform approximation of functions of bounded variation by Lagrange interpolation polynomials with a matrix ${\mathcal L}_n^{(\alpha_n,\beta_n)}$ of Jacobi nodes

Let sequences $\{\alpha_n\}_{n=1}^{\infty}$, $\{\beta_n\}_{n=1}^{\infty}$ satisfy the relations $\alpha_n\in\mathbb{R}$, $\beta_n\in\mathbb{R}$, $\alpha_n=o(\sqrt{n/\ln n})$, $\beta_n=o(\sqrt{n/\ln n})$ as $n\to \infty$, and let $[a,b]\subset (0,\pi)$ and $f\in C[a,b]$. We redefine the function $f$ as $F$ on the interval $[0,\pi]$ by polygonal arcs in such a way that the function remains continuous and vanishes on a neighbourhood of the ends of the interval. Also let the function $f$ and the pair of sequences $\{\alpha_n\}_{n=1}^{\infty}$, $\{\beta_n\}_{n=1}^{\infty}$ be connected by the equiconvergence condition. Then for the classical Lagrange–Jacobi interpolation processes $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ to approximate $f$ uniformly with respect to $\theta$ on $[a,b]$ it is sufficient that $f$ have bounded variation $V^{b}_{a}(f)<\infty$ on $[a,b]$. In particular, if the sequences $\{\alpha_n\}_{n=1}^{\infty}$ and $\{\beta_n\}_{n=1}^{\infty}$ are bounded, then for the classical Lagrange–Jacobi interpolation processes $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ to approximate $f$ uniformly with respect to $\theta$ on $[a,b]$ it is sufficient that the variation of $f$ be bounded on $[a,b]$, $V^{b}_{a}(f)<\infty$.