Lebesgue Constant (interpolation) - Sensitivity of The Values of A Polynomial

Sensitivity of The Values of A Polynomial

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