An Explicit Upper Bound for Modulus of Divided Difference on a Jordan Arc in the Complex Plane
Provides a theoretical bound for interpolation error on complex curves, relevant to approximation theory and numerical analysis.
The paper derives an explicit upper bound for the modulus of divided difference on smooth Jordan arcs/curves, providing an error estimate for complex polynomial interpolation that extends the classical result on the unit interval.
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.