三角多项式算子的饱和定理

THE SATURATION THEOREM OF TRIGONOMETRIC POLYNOMIAL OPERATORS

讨论一类三角多项式算子的饱和问题

Let f∈X p 2π , its Fourier series is S=a 02+∞k=1(a k cos kx+b k sin kx)=∞k=0A k(x), Let ( λ nk ) n,k ≥1 be a lower triangular matrix and T n(f,x)=∞k=0λ nk A k(x), where λ n 0 =1. The following theorem is obtained, Theorem Let φ n →0 + and 1< α ≤2. If the following conditions are satisfied: lim n→∞ 1- λ nk φ n=k α,(k =1,2,…); ∞k= 0 |Λ nk |(k+ 1) 1 -α =O(φ n),(n→∞), where Λ nk =λ nk - 2 λ n(k+ 1 ) +λ n(k+ 2) , then {T n} is saturated with the order O(φ n) and the saturation class F p(T n)={f∈X p 2 π :f∈ Lip (X p 2 π ,α)} in the spaces (X p 2 π ,α). In the case α =2, the theorem is obtained by the reference .