摘要
分别记n次代数多项式和x^k的系数为零的n次代数多项式对函数f∈C[a,b]的最佳逼近为E_n(f)和E_n^k(f)。1980年,M.Hasson为用f(非多项式)的光滑性来刻划E_n^k(f)/E_n(f)的有界性,提出猜想一:f∈C_([0,1])充分光滑猜想二:f∈C_([-1,1])在_([-1,1])的基一内点不可导不久前许树声否定了上述猜想.但本文证明,若将猜想二的条件加强为f除一内点a∈(-1,1)外连续可导,则结论E_n^k(f)/E_n(f)=O_((1))仍可成立。
Let E_n(f) or E_n^k(f) be the best approximation to the function f∈C_[α,b] in space π_n or π_n^k, where In 1980, M. Hasson posed the following conjectures.Conjecture 1 We have E_n^k(f)/E_n(f)=∞ if f is r-times continuously differentiable on [0,1] for r large enough.Conjecture 2 If f∈C([-1,1])and f' does not exist at some interior point of [-1,1], then E_n^k(f)/E_n(f) =O(1) (n→∞).It was shown by Xu Shusheng that both conjectures of Hasson are false. But, conjecture 2 can be true if slightly strengthen the condition of it, we prove the followingTheorem If f∈C_([1,1])(or C_([0.1]),f' is continuous except an interior point α∈[-1,1] (or[0,1]), and two one-sided derivatives f′+(a) and f′_(a) exist but f'+(α)≠f'_(α), then E_n^k(f)/E_n(f)=O(1).
出处
《数学杂志》
CSCD
北大核心
1989年第1期7-12,共6页
Journal of Mathematics