摘要
刻画具有一定逼近阶的函数空间是逼近论研究的一个重要方面.这方面的古典结果可以追溯到本世纪初D.Jackson和S.Bernstein的工作,他们证明了周期为2π的连续函数在一致范数意义下用阶为n的三角多项式来逼近时,达到阶为E_n(f)=O(n^-a),0<a<1,充要条件是f∈Lipa.对各种其它类型的逼近,包括代数多项式和等距结点样条逼近的这方面的结果也已得到.更一般地,我们考虑逼近空间A_q^a,a,q→0的刻画A_q^a={f|||f||_A_q^a<∞}。
In this paper, we establish the following Jackson and Bernstein inequalities for Cap space:S∈∑n,r,where σ=(α+ 1/p-1),α > 0,0 < p < ∞. Moreover, these inequalities are used to give characterizations of the approximation spaces resulted from two approximations verpectively by splines with free knots and approximation by rational functions.
