The Lebesgue constant can be found exactly for three nodes ( )

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

The Lebesgue constant can be found exactly for three nodes (n=2n=2). Compare two node sets on [−1,1][-1,1]: the equally spaced nodes −1,0,1-1,0,1, and the Chebyshev points of the first kind xj=cos⁡[(2j+1)π/6]x_j=\cos[(2j+1)\pi/6], j=0,1,2j=0,1,2. ((a)) For the equally spaced set, write the basis polynomials ℓj\ell_j, show that ∑j∣ℓj(x)∣=1+x−x2\sum_j|\ell_j(x)|=1+x-x^2 for 0≤x≤10\le x\le1, and find Λ2\Lambda_2 and where it occurs. ((b)) For the Chebyshev set, find ∑j∣ℓj(x)∣\sum_j|\ell_j(x)| on [0,1][0,1] in each of the two ranges into which the outer node divides it, and find Λ2\Lambda_2 and where it occurs. ((c)) Which set has the smaller Λ2\Lambda_2? Explain why this does not contradict the values Λ10=29.9\Lambda_{10}=29.9 and 2.492.49 quoted in the text for equally spaced and Chebyshev nodes.

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