An Explicit Upper Bound for Modulus of Divided Difference on a Jordan Arc in the Complex Plane

TL;DR

This paper derives an explicit upper bound for the modulus of divided differences of smooth functions on Jordan arcs or curves in the complex plane, providing improved error estimates for polynomial interpolation that are independent of curve parametrization.

Contribution

It introduces a new explicit upper bound for divided differences on Jordan curves, extending polynomial interpolation error estimates beyond the real interval.

Findings

Derived an explicit upper bound for divided differences on Jordan arcs.

Extended polynomial interpolation error estimates to complex Jordan curves.

Bound is independent of curve parametrization.

Abstract

An explicit upper bound is derived for the modulus of divided difference for a smooth(not necessarily analytic) function defined on a smooth Jordan arc (or a smooth Jordan curve) in the complex plane. As an immediate application, an error estimate for complex polynomial interpolation on a Jordan arc (or a Jordan curve) is given, which extends the well-known error estimate for polynomial interpolation on the unit interval. Moreover, this upper bound is independent of the parametrization of the curve.