- Lebesgue constant (interpolation)
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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 polynomialapproximation of the function (the degree of the polynomials are obviously fixed). The Lebesgue function for polynomials of degree at most "n" and for the set of ("n"+1) nodes "T" is generally denoted by Λ"n"("T"). These constants are named after Henri Lebesgue.
We fix the interpolation nodes "x"0, …, "x""n" and an interval ["a", "b"] 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"( ["a", "b"] ) of all continuous functions on ["a", "b"] 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 Λ"n"("T") is defined as the
operator normof "X". This definition requires us to specify a norm on "C"( ["a", "b"] ). The maximum normis usually the most convenient.
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 Π"n". Then
triangle inequality. But "X" is a projection on Π"n", so "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 Λ"n"("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 Λ"n"("T").
Minimal Lebesgue constants
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 nodesare used, since we have:where "a" = 0.9625….
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 "t""i" denote the "i"th Chebyshev node. Then, define "s""i" = "t""i"cos(π⁄2("n"+1)). For such nodes::
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