论可微函数的共单调逼近和共凸逼近

On Comonotone Approximation and Coconvex Approximation of Differentiable Functions

对有限区间上可微函数借助于代数多项式的共单调逼近和共凸逼近的逼近度估计建立了更为精确的Jackson型不等式,扩充和改进了近期的一些结果。

In this paper, by using the result about the simultaneous polynomial approximation with Hermite interpolatory side conditions, we give the Jackson type estimates for the degree of comonotone approximationEn(f)=inf{||f(x)-Pn(x)|| Pn(x)∈Πn , Pn(x) comonotone with f(x)} and coconvex approximationwhere Πn is the set of all algebraic polynomials of degree ≤n .The main results are as follows .Theorem | Let k≥2, r be any positive integers, f(x)∈Ck〔- 1〕,Y= Y = {y,|-1< <y1<.....<yr<1} be the set of points at which f'(x) = 0. If for each fixed i,1≤i ≤r, there exists positive integer ji, 2≤ji≤k, such thatf(j)(yi) = 0, j = 1,…,ji-1; f(ji)(yi)≠0,then, for all n sufficiently large,where C only depends on Y.Theorem 2 Let k≥3, s be any positive integers, f(x)∈Ck〔C- 1,1〕, Z={z,|-1 <z1<.....<zs<1} be the set of points at which f'(x) = 0.If for each fixed i, 1≤i≤ s, there exists positive integer ji , 3≤ji≤k, such thatthen , for all R sufficiently large,where C only depends on Z .