Lebesgue functions and Lebesgue constants in polynomial interpolation

The Lebesgue constant is a valuable numerical instrument for linear interpolation because it provides a measure of how close the interpolant of a function is to the best polynomial approximant of the function. Moreover, if the interpolant is computed by using the Lagrange basis, then the Lebesgue constant also expresses the conditioning of the interpolation problem. In addition, many publications have been devoted to the search for optimal interpolation points in the sense that these points lead to a minimal Lebesgue constant for the interpolation problems on the interval [–1, 1]. In Section 1 we introduce the univariate polynomial interpolation problem, for which we give two useful error formulas. The conditioning of polynomial interpolation is discussed in Section 2. A review of some results for the Lebesgue constants and the behavior of the Lebesgue functions in view of the optimal interpolation points is given in Section 3.