反函数的混合多项式逼近

Blended polynomial approximations to the inverse.

给定一个在[0,α]上的单调函数λ=f(t),给出一个函数序列pm(λ)来逼近其反函数t=f-1(λ),其中pm(λ)不是一般的多项式函数,而是多项式和三角函数的混合,即pm(λ)∈Ωm=span{sint,cost,1,t,t2,…,tm-2},称这样的逼近为混合多项式逼近.利用Ωm中有一组标准正交基,即拟Legendre基,可以表示出pm(λ).通过比较可得,对于一些特定的函数,混合多项式逼近比以往的多项式逼近效果要好.

Let λ= f（t） be a monotone function on [0,α], then a sequence of approximations Pm（λ） to the inverse t： f^-1（λ） is computed, where Pm（λ） is not a polynomial function, but a function of the blend of a polynomial function and a trigonometric function, that is Pm（λ）∈Ωm=span{sint,cos t,1,t,t^2,….t^（m-2）}. So the approximations are called blended polynomial approximations, Pm（λ） can be denoted as the standardized orthogonal basis of Ωm. Compared to the polynomial approximations, the blended polynomial approximations are more close to some of the inverse functions.