Two results on polynomial interpolation in equally spaced points
By L. N. Trefethen, J. Andre Weideman, C. K. Chui. 1991
Abstract We present two results that quantify the poor behavior of polynomial interpolation in n equally spaced points. First, in band-limited interpolation of complex exponential functions e iαx (α ϵ R ), the error decreases to 0 as n → ∞ if and only if α is small enough to provide at least six points per wavelength. Second, the Lebesgue constant Λ n (supremum norm of the n th interpolation operator) satisfies lim n → ∞ Λ n 1 n = 2 . Both of these results are more than 50 years old, but they are generally unknown to approximation theorists.
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Citations in the corpus
- Fast Interpolation-based t-SNE for Improved Visualization of Single-Cell RNA-Seq Data (2017, background): “Evidently, Hence A slight modification of the argument presented in Trefethen and Weideman (1991) 21 yields the following bound, from which it follows that which is the estimate we require.”
- Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-Periodic Functions, Part I: Single-Interval Schemes (2009, background): “Another is that the interpolation matrix on an equispaced grid is very ill-conditioned [35].”