For interpolation nodes on the Lebesgue constant obeys

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Approximating a function: interpolation and differentiation

Problem

For n+1n+1 interpolation nodes on [−1,1][-1,1] the Lebesgue constant obeys, for equally spaced nodes (an asymptotic estimate) and for Chebyshev nodes (an upper bound) respectively, Λn≈2n+1e nln⁡n,Λn≤2πln⁡(n+1)+1.\begin{aligned}\Lambda_n &\approx \frac{2^{n+1}}{e\,n\ln n},\\ \Lambda_n &\le \frac{2}{\pi}\ln(n+1) + 1.\end{aligned} A spectrometer trace is to be modelled by a degree-15 polynomial (n=15n = 15, so 1616 nodes) through samples taken either at equally spaced wavelengths or at Chebyshev-spaced wavelengths on the same interval. ((a)) Estimate or bound Λ15\Lambda_{15} for each node set from the formulas above. ((b)) If both interpolants have the same best-approximation error ∥f−p∗∥\|f - p^\ast\|, the error bound is (1+Λn)∥f−p∗∥(1+\Lambda_n)\|f-p^\ast\|. By roughly what factor is this bound larger for the equally spaced samples than for the Chebyshev samples?

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