On the Stability of Interpolation

Abstract

Some definitions on stability of interpolating process are given and then the sufficient and necessary conditions are obtained. On this basis, we conclude that the Lagrange interpolation is unstable, whereas several types of piecewise low order polynomial interpolation are stable. For high order approximation with data on isometric nodes, we recommend the Bernstein approximation owing to its high stability. Some ideas on the relationship between stability and convergence of interpolating process are also presented.