Properties
The Lebesgue constant bounds the interpolation error:
We will here prove this statement with the maximum norm. Let p∗ denote the best approximation of f among the polynomials of degree n or less. In other words, p∗ minimizes ||p − f|| among all p in Πn. Then
by the triangle inequality. But X is a projection on Πn, so p∗ − X(f) =X(p∗) − X(f) = X(p∗−f). This finishes the proof. 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.
The Lebesgue constant can be expressed in terms of the Lagrange basis polynomials:
In fact, we have the Lebesgue function
and the Lebesgue constant (or Lebesgue number) for the grid is its maximum value
Nevertheless, it is not easy to find an explicit expression for Λn(T).
Read more about this topic: Lebesgue Constant (interpolation)
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