Divergence (Runge Phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock-Chebyshev subset interpolation

摘要

Runge showed more than a century ago that polynomial interpolation of a function f(х),using points evenly spaced on x E [-1,1], could diverge on parts of this interval even if f(х) was analytic everywhere on the interval. Least-squares fitting of a polynomial of degree N to an evenly spaced grid with P points should improve accuracy if P_ N. We show through numerical experiments that such an overdetermined fit reduces but does not elim_inate the Runge Bhenomenon. More precisely, define _ = (N + 1)/P. The least-squares fit will fail to converge everywhere on [-1,1) as N ooh for fixed