# On one estimate of divided differences and its applications

## TL;DR

This paper provides an estimate for divided differences with coalescing points, strengthening classical inequalities in Hermite interpolation and deriving new approximation bounds for smooth functions.

## Contribution

It introduces a novel estimate for divided differences with coalescing points and applies it to improve Whitney and Marchaud inequalities in Hermite interpolation.

## Key findings

- Strengthened bounds for Hermite interpolation errors.
- Derived approximation estimates involving moduli of smoothness.
- Improved understanding of divided differences with multiple coalescing points.

## Abstract

We give an estimate of the general divided differences $[x_0,\dots,x_m;f]$, where some of the $x_i$'s are allowed to coalesce (in which case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen Whitney and Marchaud celebrated inequalities in relation to Hermite interpolation. For example, one of the numerous corollaries of this estimate is the fact that, given a function $f\in C^{(r)}(I)$ and a set $Z=\{z_j\}_{j=0}^\mu$ such that $z_{j+1}-z_j \geq \lambda |I|$, for all $0\le j \le \mu-1$, where $I:=[z_0, z_\mu]$, $|I|$ is the length of $I$ and $\lambda$ is some positive number, the Hermite polynomial ${\mathcal L}(\cdot;f;Z)$ of degree $\le r\mu+\mu+r$ satisfying ${\mathcal L}^{(j)}(z_\nu; f;Z) = f^{(j)}(z_\nu)$, for all $0\le \nu \le \mu$ and $0\le j\le r$, approximates $f$ so that, for all $x\in I$, \[ \big|f(x)- {\mathcal L}(x;f;Z) \big| \le C \left( \mathop{\rm dist}\nolimits(x, Z) \right)^{r+1} \int_{\mathop{\rm dist}\nolimits(x, Z)}^{2|I|}\frac{\omega_{m-r}(f^{(r)},t,I)}{t^2}dt , \] where $m :=(r+1)(\mu+1)$, $C=C(m, \lambda)$ and $\mathop{\rm dist}\nolimits(x, Z) := \min_{0\le j \le \mu} |x-z_j|$.