In numerical analysis, 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 {\displaystyle n} and for the set of n + 1 {\displaystyle n+1} nodes T {\displaystyle T} is generally denoted by Λ n ( T ) {\displaystyle \Lambda _{n}(T)} . These constants are named after Henri Lebesgue.
Definition We fix the interpolation nodes x 0 , . . . , x n {\displaystyle x_{0},...,x_{n}} and an interval [ a , b ] {\displaystyle [a,\,b]} containing all the interpolation nodes. The process of interpolation maps the function f {\displaystyle f} to a polynomial p {\displaystyle p} . This defines a mapping X {\displaystyle X} from the space C ( [ a , b ] ) {\displaystyle C([a,b])} of all continuous functions on [ a , b ] {\displaystyle [a,b]} to itself. The map X {\displaystyle X} is linear and it is a projection on the subspace Πn of polynomials of degree n or less. The Lebesgue constant Λ n ( X ) {\displaystyle \Lambda _{n}(X)} is defined as the operator norm of X {\displaystyle X} . This definition requires us to specify a norm on C ( [ a , b ] ) {\displaystyle C([a,b])} . The uniform norm is usually the most convenient.
Properties The Lebesgue constant bounds the interpolation error: let p ∗ {\displaystyle p^{*}} denote the best approximation of f {\displaystyle f} among the polynomials of degree n {\displaystyle n} or less. In other words, p ∗ {\displaystyle p^{*}} minimizes ‖ p − f ‖ {\displaystyle \|p-f\|} among all p in Πn. Then
‖ f − X ( f ) ‖ ≤ ( Λ n ( T ) + 1 ) ‖ f − p ∗ ‖ . {\displaystyle \|f-X(f)\|\leq (\Lambda _{n}(T)+1)\left\|f-p^{*}\right\|.}
We will here prove this statement with the maximum norm.
‖ f − X ( f ) ‖ ≤ ‖ f − p ∗ ‖ + ‖ p ∗ − X ( f ) ‖ {\displaystyle \|f-X(f)\|\leq \|f-p^{*}\|+\|p^{*}-X(f)\|}
by the triangle inequality. But X {\displaystyle X} is a projection on Πn, so
p∗ − X( f ) = X(p∗) − X( f ) = X(p∗ − f ). This finishes the proof since ‖ X ( p ∗ − f ) ‖ ≤ ‖ X ‖ ‖ p ∗ − f ‖ = ‖ X ‖ ‖ f − p ∗ ‖ {\displaystyle \|X(p^{*}-f)\|\leq \|X\|\|p^{*}-f\|=\|X\|\|f-p^{*}\|} . 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:
ℓ j ( x ) := ∏ i = 0 j ≠ i n x − x i x j − x i . {\displaystyle \ell _{j}(x):=\prod _{\begin{smallmatrix}i=0\\j\neq i\end{smallmatrix}}^{n}{\frac {x-x_{i}}{x_{j}-x_{i}}}.}
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