In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree of the polynomials are fixed).
The Lebesgue constant for polynomials of degree at most n and for the set of n + 1 nodes T is generally denoted by Λn(T ).
The map X is linear and it is a projection on the subspace Πn of polynomials of degree n or less.
The Lebesgue constant bounds the interpolation error: let p∗ denote the best approximation of f among the polynomials of degree n or less.
Note that this relation comes also as a special case of Lebesgue's lemma.
In other words, the interpolation polynomial is at most a factor Λn(T ) + 1 worse than the best possible approximation.
This suggests that we look for a set of interpolation nodes with a small Lebesgue constant.
In the case of equidistant nodes, the Lebesgue constant grows exponentially.
More precisely, we have the following asymptotic estimate On the other hand, the Lebesgue constant grows only logarithmically if Chebyshev nodes are used, since we have We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation.
However, there is an easy (linear) transformation of Chebyshev nodes that gives a better Lebesgue constant.
Let ti denote the i-th Chebyshev node.
the set of n times differentiable functions whose n-th derivatives are bounded in absolute values by a constant M as shown by N. S. Hoang.
Using a computer, one can approximate the values of the minimal Lebesgue constants, here for the canonical interval [−1, 1]: There are uncountable infinitely many sets of nodes in [−1,1] that minimize, for fixed n > 1, the Lebesgue constant.
All arbitrary (i.e. zero-symmetric or zero-asymmetric) optimal sets of nodes in [−1,1] when n = 2 have been determined by F. Schurer, and in an alternative fashion by H.-J.
Rack (1984 and 2013), for the case n = 3, the explicit values of the optimal (unique and zero-symmetric) 4 interpolation nodes and the explicit value of the minimal Lebesgue constant are known.
All arbitrary optimal sets of 4 interpolation nodes in [1,1] when n = 3 have been explicitly determined, in two different but equivalent fashions, by H.-J.
The Padua points provide another set of nodes with slow growth (although not as slow as the Chebyshev nodes) and with the additional property of being a unisolvent point set.
Consider the inequality: This means that the (relative) error in the values of
will not be higher than the appropriate Lebesgue constant times the relative error in the coefficients.
In this sense, the Lebesgue constant can be viewed as the relative condition number of the operator mapping each coefficient vector u to the set of the values of the polynomial with coefficients u in the Lagrange form.
We can actually define such an operator for each polynomial basis but its condition number is greater than the optimal Lebesgue constant for most convenient bases.