Limitations of Chebyshev interpolation nodes : Counter examples and Insights

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.