H_q^p(p>0,q>1)空间多项式最佳逼近的一些结果

Some Results on Best Approximation by Polynomials in H_q^p(p>0,q>1) Spaces

本文在Hpq (p> 0, q> 1) 空间中证明了伯恩斯坦(Bernstein) 型不等式, 从而得到了关于多项式最佳逼近阶的估计的逆定理.

A Berntein type inequality and a converse theorem of best approximation by polynomials in H p q(p>0,q>1) spaces are proved. The main results are Theorem 1.For any polymomial P n(z) of degreen and p>0,q>1,we have‖P ′ n(z)‖≤c pnr 0‖P n(z)‖,where r 0=min12,12｜2q-3｜ -1/2 ,c p is a constant depending only on p.Particularly, if p≥1,c p can be taken as 4. ,c p is a constant depending only on p.Particularly, if p≥1,c p can be taken as 4. Theorem 2. Let f(z) be a function defined in the unit disc ｜z｜<1, the function Ω(u) nondecreasing in u>0 and the integral∫ t 0 Ω μ(u)uduconvergent, where μ=min｛p,1｝. Suppose that there exists a constant A such that Ω(2u) ≤AΩ(u) and for any positive integer n there exists a polynomial P 2 n (z) of degree 2 n such that‖f(z)-p 2 n (z)‖≤B2 nm Ω(12 n),p>0,q>1,where m is a non negative integer, B is a constant independent of n. Then we havef (υ) (z)∈H p q(p>0,q>1),υ=0,1,…,m,andω(t,f (m) ≤Ct μ∫ 1 t Ω μ(u)u μ+1 du+∫ t 0 Ω μ(u)uduΩ μ(u)uduconvergent, where μ=min｛p,1｝. Suppose that there exists a constant A such that Ω(2u) ≤AΩ(u) and for any positive integer n there exists a polynomial P 2 n (z) of degree 2 n such that‖f(z)-p 2 n (z)‖≤B2 nm Ω(12 n),p>0,q>1,where m is a non negative integer, B is a constant independent of n. Then we havef (υ) (z)∈H p q(p>0,q>1),υ=0,1,…,m,andω(t,f (m) ≤Ct μ∫ 1 t Ω μ(u)u μ+1 du+∫ t 0 Ω μ(u)uduuduconvergent, where μ=min｛p,1｝. Suppose that there exists a constant A such that Ω(2u) ≤AΩ(u) and for any positive integer n there exists a polynomial P 2 n (z) of degree 2 n such that‖f(z)-p 2 n (z)‖≤B2 nm Ω(12 n),p>0,q>1,where m is a non negative integer, B is a constant independent of n. Then we havef (υ) (z)∈H p q(p>0,q>1),υ=0,1,…,m,andω(t,f (m) ≤Ct μ∫ 1 t Ω μ(u)u μ+1 du+∫ t 0 Ω μ(u)ududuconvergent, where μ=min｛p,1｝. Suppose that there exists a constant A such that Ω(2u) ≤AΩ(u) and for any positive integer n there exists a polynomial P 2 n (z) of degree 2 n such that‖f(z)-p 2 n (z)‖≤B2 nm Ω(12 n),p>0,q>1,where m is a non negative integer, B is a constant independent of n. Then we havef (υ) (z)∈H p q(p>0,q>1),υ=0,1,…,m,andω(t,f (m) ≤Ct μ∫ 1 t Ω μ(u)u μ+1 du+∫ t 0 Ω μ(u)udumin｛p,1｝. Suppose that there exists a constant A such that Ω(2u) ≤AΩ(u) and for any positive integer n there exists a polynomial P 2 n (z) of degree 2 n such that‖f(z)-p 2 n (z)‖≤B2 nm Ω(12 n),p>0,q>1,where m is a non negative integer, B is a constant independent of n. Then we havef (υ) (z)∈H p q(p>0,q>1),υ=0,1,…,m,andω(t,f (m) ≤Ct μ∫ 1 t Ω μ(u)u μ+1 du+∫ t 0 Ω μ(u)udu｛p,1｝. Suppose that there exists a constant A such that Ω(2u) ≤AΩ(u) and for any positive integer n there exists a polynomial P 2 n (z) of degree 2 n such that‖f(z)-p 2 n (z)‖≤B2 nm Ω(12 n),p>0,q>1,where m is a non negative integer, B is a constant independent of n. Then we havef (υ) (z)∈H p q(p>0,q>1),υ=0,1,…,m,andω(t,f (m) ≤Ct μ∫ 1 t Ω μ(u)u μ+1 du+∫ t 0 Ω μ(u)uduB2 nm Ω(12 n),p>0,q>1,where m is a non negative integer, B is a constant independent of n. Then we havef (υ) (z)∈H p q(p>0,q>1),υ=0,1,…,m,andω(t,f (m) ≤Ct μ∫ 1 t Ω μ(u)u μ+1 du+∫ t 0 Ω μ(u)udu2 nm Ω(12 n),p>0,q>1,where m is a non negative integer, B is a constant independent of n. Then we havef (υ) (z)∈H p q(p>0,q>1),υ=0,1,…,m,andω(t,f (m) ≤Ct μ∫ 1 t Ω μ(u)u μ+1 du+∫ t 0 Ω μ(u)uduΩ(12 n),p>0,q>1,where m is a non negative integer, B is a constant independent of n. Then we havef (υ) (z)