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This work focuses on weighted Lagrange interpolation on an unbounded domain and analyzes the Lebesgue constant for a sequence of weighted Leja points. The standard Leja points are a nested sequence of points defined on a compact subset of the real line and can be extended to unbounded domains with the introduction of a weight function w: R→[0,1]. Due to a simple recursive formulation in one dimension, such abscissas provide a foundation for high-dimensional approximation methods such as sparse grid collocation, deterministic least squares and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes.
Figure: This is the main theorem of this work. We explore the connection between polynomials and weighted potentials, and show how classical weighted potential theory can be used to understand the asymptotic behavior (with respect to n) of an nth degree polynomial with roots at the contracted Leja points. While these techniques give us most of the result, the final part of the proof requires an explicit estimate on the spacing of the weighted Leja nodes.
Publication:
Publication:
P. Jantsch, C. Webster and G. Zhang, On the Lebesgue constants of weighted Leja points for Lagrange interpolation on unbounded domains, IMA Journal on Numerical Analysis, 39 (2), 1039-1057, 2019.
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