Limitations of Chebyshev interpolation nodes : Counter examples and Insights

El-Malki, Imane; Alaoui, Mounia; Mtai, Hanaa; Laghrib, Amine

doi:10.47974/JIM-1793

Open Access Research Article

Limitations of Chebyshev interpolation nodes : Counter examples and Insights

Imane El-MalkiEMI Béni-Mellal Université Sultan Moulay Slimane Campus Mghilla BP 523, 23000Béni Mellal, MoroccoView full profile → , Mounia AlaouiEMI Béni-Mellal Université Sultan Moulay Slimane Campus Mghilla BP 523, 23000Béni Mellal, MoroccoView full profile → , Hanaa MtaiEMI Béni-Mellal Université Sultan Moulay Slimane Campus Mghilla BP 523, 23000 Béni Mellal, MoroccoView full profile → , ^*Amine LaghribCorresponding authorlaghrib.amine@gmail.comEMI Béni-Mellal Université Sultan Moulay Slimane Campus Mghilla BP 523, 23000 Béni Mellal, 40000, MoroccoView full profile →

* Corresponding author · click or hover a name for details

pp. 575–600Vol. 27Issue 3April 2024DOI: 10.47974/JIM-1793XML

Received:: 07 Jun 2023
Published Online:: 07 May 2024
Article type:: Research Article
Language:: EN
Article no.:: JIM-1793
Pages:: 575–600

Abstract

Interpolation using uniformly spaced nodes often encounters Runge’s phenomenon when applied to smooth functions. To mitigate this issue, Chebyshev roots are frequently recommended as better interpolation nodes. However, our study reveals the limitations of Chebyshev roots for interpolation by presenting a series of counterexamples. These counterexamples encompass functions that exhibit both differentiable and non-differentiable characteristics. By examining scenarios where Chebyshev interpolation falls short of producing satisfactory results, we shed light on when alternative interpolation methods should be considered.

Keywords

Runge phenomenon Non-differentiable Non-linear function Lagrange approximation Chebyshev nodes

Subject Classifications

Primary 49M05Secondary 53C35

References

[1] L. N. Trefethen. Approximation Theory and Approximation Practice, Extended Edition. SIAM (2019).
[2] S. Roberts, A. Sarshar and A. Sandu. Parallel implicit-explicit general linear methods. Communications on Applied Mathematics and Computation, 3(4) : 649-669 (2019).
[3] J. P. Boyd and J. R. Ong. Exponentially-convergent strategies for defeating the Runge Phenomenon for the approximation of non-periodic functions, part two: Multi-interval polynomial schemes and multidomain Chebyshev interpolation. Applied Numerical Mathematics, 61(4) : 460-472 (2011).
[4] J. P. Boyd and J. R. Ong. Whitney edge elements and the Runge phenomenon. Journal of Computational and Applied Mathematics, 427(1) : 115-127 (2023).
[5] E. Cheney. Chebyshev methods in numerical approximation (martin avery snyder). SIAM Review, 9(2) : 264 (1967).
[6] S. Jawad Jaaywel, K. Hashim Abdul-Khaliq. The error analysis of linearized discontinuous Galerkin finite element method for incompressible miscible displacement in porous media. Journal of Interdisciplinary Mathematics, 22(8) : 1471-1484 (2019).
[7] J. Jabbar Hussein, K. Hashim Abdul-Khaliq. A weak Galerkin finite element method for two-dimensional coupled burgersâ€™ equation by using polynomials of order (k, k-1, k-1). Journal of Interdisciplinary Mathematics, 23(4) : 777-790 (2020).
[8] A. Laghrib, A. Hadri, M. Hakim. An Enhanced Fluid Registration for Image Multi-Frame Super Resolution. Scientific Annals of Computer Science, 32(2):11-34 (2022).
[9] M. Alahyane, A. Hakim, A. Laghrib, S. Raghay. An improved SIMPLEC scheme for fluid registration. Mathematical Modelling and Analysis, 28(1) : 71-90 (2023).
[10] B. Fornberg and J. Zuev. The runge phenomenon and spatially variable shape parameters in rbf interpolation. Computers & Mathematics with Applications, 54(3) : 379-398 (2007).
[11] J. C. Mason and D. C. Handscomb. Chebyshev polynomials. Chapman and Hall/CRC (2002).

Views: 238Downloads: 9Citations: 1