Polynomial Interpolation - Lebesgue Constants

Lebesgue Constants

See the main article: Lebesgue constant.

We fix the interpolation nodes x₀, ..., x_n and an interval containing all the interpolation nodes. The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C of all continuous functions on to itself. The map X is linear and it is a projection on the subspace Π_n of polynomials of degree n or less.

The Lebesgue constant L is defined as the operator norm of X. One has (a special case of Lebesgue's lemma):

In other words, the interpolation polynomial is at most a factor (L + 1) worse than the best possible approximation. This suggests that we look for a set of interpolation nodes that L small. In particular, we have for Chebyshev nodes:

We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation, as the growth in n is exponential for equidistant nodes. However, those nodes are not optimal.