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
Abstract:
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
$[a,b]$, $V^{b}_{a}(f)<\infty$.
Keywords:sinc-approximations, interpolation of functions, uniform approximation, interpolation polynomials, bounded variation.
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