The Trigonometric And Paratrigonometric Rational Interpolations Of Barycentric Form

In this thesis, the study for the barycentric formulas of Lagrange trigonometric and paratrigonometric interpolation with free nodes on [ 0,2π), the barycentric rational function and the derivatives of barycentric rational interpolation are discussed. The full text is divided into four parts.In the first part, the definitions of the 2π-periodic function, 2π-antiper- iodic function, the trigonometrc and paratricgonometric polynomials and their familis are introduced. Some of basic properties of the trigonometrc and paratrigonometric polynomial in the family are discussed.In the second part, the problems of the barycentric form of Lagrange trigonometric interpolation in the class and its derivatives are studied. At first, the barycentric form of Lagrange trigonometric interpolation in the family and its trigonometric rational interpolation function are given. Then we prove that the function has no pole on the real line. Furthermore, a recurrence relation for calculating the p^th differentiation matrix which is needed by the derivatives on the nodes and an iterative formula for computing the p^thderivative off the nodes of the barycentric trigonometric rational interpolation are given. The results of Salzer, Berrut and Baltensperger are special cases of this thesis.In the third part, we discuss the barycentric form and its derivatives of Lagrange paratrigonometric interpolation. The barycentric formula of Lagrange paratrigonometric interpolation in the class is proposed. The recurrence formula of the p^th differentiation matrix and an iterative relation of p^th derivative off the nodes for the paratri- gonometric rational interpolation of barycentric form are obtained. The fourth part will focus on the stability and the convergence of the barycentric rational interpolation. We generalize the results of Baltensperger and Berrut to trigonometric rational interpolation.