加权K-泛函和Fourier-Chebyshev展开的Riesz型平均( 英文)

A Weighted K-functional and the Riesz-Type Means from Fourier-Chebyshev Expansion

对 1 p +∞ ,r∈N+,建立了下列等价关系 :‖w(R(T)n -I) rf‖ p～K2r(f ,n- 2r) w ,p～‖w(R(T)n ,rf - f)‖p,其中权函数w(x) =(1-x2 ) - 12 p,R(T)n ,r(f ,x) =∑nk =0 (1- k2rn2r)ak(f)Tk(x)是函数 f的Fourier Chebyshev展开的r阶Riesz型平均 ,R(T)n =(f ,x) =R(T)n ,1(f ,x) ,K2r(f ,tr) w ,p是一个K 泛函 ,定义为 :K2r(f ,tr) w ,p=infg∈C2r[-1,1](‖w(f - g)‖p+tr‖wP(D) rg‖p) 。

For 1p+∞ and r∈N +, The following equivalent relationship is established. ‖w(R (T) n-I) rf‖ p～K 2r (f,n -2r ) w,p ～‖w(R (T) n,r f-f)‖ p, where the weight function w(x)=(1-x 2) -12p , R (T) n,r (f,x)=∑nk=0(1-k 2r n 2r )a k(f)T k(x), is the Riesz type means of order r of Fourier Chebyshev expansion of f and K 2r (f,t r) w,p is a new weighted K functional defined by K 2r (f,t r) w,p = inf g∈C 2r [-1,1] (‖w(f-g)‖ p+t r‖wP(D) rg‖ p), Where the differential operator P(D)=1-x 2ddx1-x 2ddx .