A barycentric trigonometric Hermite interpolant via an iterative approach

TL;DR

This paper develops a new barycentric rational trigonometric Hermite interpolant using an iterative approach, extending previous methods to improve computational efficiency and convergence analysis for interpolation at various nodes.

Contribution

It generalizes an iterative scheme for Hermite interpolation, applying it to barycentric rational trigonometric interpolants with analytical differentiation matrices.

Findings

Effective construction of the interpolant demonstrated

Convergence rate analyzed at different node distributions

Numerical examples validate the method's accuracy

Abstract

In this work we construct an Hermite interpolant starting from basis functions that satisfy a Lagrange property. In fact, we extend and generalise an iterative approach, introduced by Cirillo and Hormann (2018) for the Floater-Hormann family of interpolants. Secondly, we apply this scheme to produce an effective barycentric rational trigonometric Hermite interpolant at general ordered nodes using as basis functions the ones of the trigonometric interpolant introduced by Berrut (1988). For an easy computational construction, we calculate analytically the differentation matrix. Finally, we conclude with various examples and a numerical study of the rate of convergence at equidistant nodes and conformally mapped nodes.