Lebesgue constants

I alluded to Lebesgue constants in the previous post without giving them a name. There I said that the bound on order ninterpolation error has the form

whereh is the spacing between interpolation points and is the error in the tabulated values. The constantc depends on the function f being interpolated, and to a lesser extent onn. The constant is independent off but depends onn and on the relative spacing between the interpolation nodes. This post will look closer at .

Given a set of n + 1 nodes T

define

Then the Lebesgue function is defined by

and the Lebesgue constant for the grid is the maximum value of the Lebesgue function

The values of are difficult to compute, but there are nice asymptotic expressions for when the grid is evenly spaced:

When the grid points are at the roots of a Chebyshev polynomial then

The previous post mentioned the casesn = 11 andn = 29 for evenly spaced grids. The corresponding values of are approximately 155 and 10995642. So 11th order interpolation is amplifying the rounding error in the tabulated points by a factor of 155, which might be acceptable. But 29th order interpolation is amplifying the rounding error by a factor of over 10 million.

The corresponding values of for Chebyshev-spaced nodes are 2.58 and 3.17. Chebyshev spacing is clearly better for high-order interpolation, when you have that option.

The post Lebesgue constants first appeared on John D. Cook.