On the uniform approximation of functions of bounded variation by Lagrange interpolationpolynomials 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 therelations $\alpha_n\in\mathbb{R}$, $\beta_n\in\mathbb{R}$,$\alpha_n=o(\sqrt{n/\ln n})$, $\beta_n=o(\sqrt{n/\ln n})$ 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 $[0,\pi]$ bypolygonal 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 $f$ andthe pair of sequences $\{\alpha_n\}_{n=1}^{\infty}$, $\{\beta_n\}_{n=1}^{\infty}$ beconnected by the equiconvergence condition. Then for the classical Lagrange–Jacobiinterpolation processes $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ toapproximate $f$uniformly with respect to $\theta $ on $[a,b]$ it is sufficient that $f$ have bounded variation $V^{b}_{a}(f)<\infty$ on $[a,b]$. Inparticular, if the sequences $\{\alpha_n\}_{n=1}^{\infty}$ and$\{\beta_n\}_{n=1}^{\infty}$ are bounded, then for the classical Lagrange–Jacobiinterpolation processes $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ toapproximate $f$ uniformly with respect to $\theta $on $[a,b]$ it is sufficient that the variation of $f$ be bounded on$[a,b]$, $V^{b}_{a}(f)<\infty$.

sinc-approximations, interpolation of functions, uniform approximation, interpolation polynomials, bounded variation