𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On Generalized Hermite–Fejér Interpolation of Lagrange Type on the Chebyshev Nodes

✍ Scribed by Graeme J. Byrne; T.M. Mills; Simon J. Smith

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
131 KB
Volume
105
Category
Article
ISSN
0021-9045

No coin nor oath required. For personal study only.

✦ Synopsis

For f # C [&1, 1], let H m, n ( f, x) denote the (0, 1, ..., m) Hermite Feje r (HF) interpolation polynomial of f based on the Chebyshev nodes. That is, H m, n ( f, x) is the polynomial of least degree which interpolates f (x) and has its first m derivatives vanish at each of the zeros of the nth Chebyshev polynomial of the first kind. In this paper a precise pointwise estimate for the approximation error |H 2m, n ( f, x) & f (x)| is developed, and an equiconvergence result for Lagrange and (0, 1, ..., 2m) HF interpolation on the Chebyshev nodes is obtained. This equiconvergence result is then used to show that a rational interpolatory process, obtained by combining the divergent Lagrange and (0, 1, ..., 2m) HF interpolation methods on the Chebyshev nodes, is convergent for all

m, n (X, f, x k, n )=0, 1 k n, 1 r m, 1 k n.

📜 SIMILAR VOLUMES

On the Positivity of the Fundamental Pol
On the Positivity of the Fundamental Polynomials for Generalized Hermite–Fejér Interpolation on the Chebyshev Nodes
✍ Simon J. Smith 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 90 KB

It is shown that the fundamental polynomials for (0, 1, ..., 2m+1) Hermite Feje r interpolation on the zeros of the Chebyshev polynomials of the first kind are nonnegative for &1 x 1, thereby generalising a well-known property of the original Hermite Feje r interpolation method. As an application of

On Normal Pointsystems of Hermite–Fejér
On Normal Pointsystems of Hermite–Fejér Interpolation of Arbitrary Order
✍ Ying Guang Shi 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 127 KB

Necessary conditions of normal pointsystems for Hermite-Fejér interpolation of arbitrary (even) order are given. In particular, one of the main results in this paper is: If a pointsystem consists of the zeros of orthogonal polynomials with respect to a weight w on [-1, 1] and is always normal for He

Uniform Convergence of Lagrange Interpol
Uniform Convergence of Lagrange Interpolation Based on the Jacobi Nodes
✍ George Kvernadze 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 485 KB

Necessary and sufficient conditions are obtained for a continuous function guaranteeing the uniform convergence on the whole interval [ &1, 1] of its Lagrange interpolant based on the Jacobi nodes. The conditions are in terms of 4-variation, 8-variation, the modulus of variation, and the Banach indi