Lebesgue constants in polynomial interpolation

Lagrange interpolation is a classical method for approximating a continuous function by a polynomial that agrees with the function at a number of chosen points (the “nodes”). However, the accuracy of the approximation is greatly influenced by the location of these nodes. Now, a useful way to measure a given set of nodes to determine whether its Lagrange polynomials are likely to provide good approximations is by means of the Lebesgue constant. In this paper a brief survey of methods and results for the calculation of Lebesgue constants for some particular node systems is presented. These ideas are then discussed in the context of Hermite–Fejér interpolation and a weighted interpolation method where the nodes are zeros of Chebyshev polynomials of the second kind.