Some Thoughts on the Polynomial Approximations to a Given Function

Yih-Yuh Chen

Chinese Journal of Physics

38卷5期（2000 / 10 / 01）

927 - 938

英文

Traditionally, the polynomial approximation to a given function can be conveniently achieved using a Taylor expansion, provided the function is sufficiently differentiable. More generally, one can use Bernstein's polynomials to approximate any continuous function defined on a closed interval. We show that minor modifications to Bernstein's polynomials yield an even smaller set of polynomial functions that can explicitly and faithfully approximate a given function. The Gibbs phenomenon associated with this approximation is shown to result from the universality of the approximation near a discontinuity when certain limits are taken.