Hermite Interpolation Using ERBS with Trigonometric Polynomial Local Functions

Abstract

Given a sequence of knots t ₀ < t ₁ < ⋯ < t _n + 1, an expo-rational B-spline (ERBS) function f(t) is defined by

$$ f(t)=\sum_{k=1}^n \ell_k(t)B_k(t), \quad t\in[t_1,t_n], $$

where B _k (t) are the ERBS and ℓ _k (t) are local functions defined on (t _k − 1,t _k + 1). Consider the Hermite interpolation problem at the knots 0 ≤ t ₁ < t ₂ < ⋯ < t _n  < 2π of arbitrary multiplicities. In [3] a formula was suggested for Hermite-interpolating ERBS function with ℓ _k (t) being algebraic polynomials. Here we construct Hermite interpolation by an ERBS function with trigonometric polynomial local functions. We provide also numerical results for the performance of the new trigonometric ERBS (TERBS) interpolant in graphical comparison with the interpolant from [3]. Potential applications and some topics for further research on TERBS are briefly outlined.

The research of Dechevsky and Uluchev was partially supported, respectively, by the 2010 and 2011 Annual Research Grant of the Priority R&D Group for Mathematical Modeling, Numerical Simulation and Computer Visualization at Narvik University College, and by the Bulgarian Ministry of Education, Youth and Science under Grant No. DDVU 02/30, 2010.