Лебег тұрақтылары: интерполяция және жақындық сапасы

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 ). These constants are named after Henri Lebesgue.

We fix the interpolation nodes and an interval containing all the interpolation nodes. The process of interpolation maps the function to a polynomial This defines a mapping 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 Πn of polynomials of degree n or less. The Lebesgue constant is defined as the operator norm of X. This definition requires us to specify a norm on C([a, b]). The uniform norm is usually the most convenient.

The Lebesgue constants also arise in another problem. Let p(x) be a polynomial of degree n expressed in the Lagrangian form associated with the points in the vector t (i. e. the vector u of its coefficients is the vector containing the values ). Let be a polynomial obtained by slightly changing the coefficients u of the original polynomial p(x) to 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.