加权K-泛函和Fourier-Chebyshev展开的Riesz型平均

对1≤p≤ ∞，r∈N’≤^ ，建立了下列等价关系：||w(Rn^(T)-I)f||p～K2r(f,n^-2r)w,p～||w(Rn(T)r,f-f)||p,其中权函数w(x)=(2-x^2)^-(1/2p),Rn(T)r(f,x)=^n∑k=0(1-(k^2r/n^2r))ak(f)Tk(x)是函数f的Fourier-Chebyshev展开的r阶Riesz型平均，Rn(T)=(f,x)=Rn^(T),1(f,x),K2r(f,t^r)w,p是一个K-泛函，定义为：K2r(f,t')w,p=(^g∈C^2r-1,1])inf (||w(f-g)||p t'||wP(D)'g||p),这里微分算式P（D）=√1-x^2(d/dx)√1-x^2(d/dx).

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

For 1 ≤p ≤ + 8 and r ∈ N + , The following equivalent relationship is established.‖ w( Rn(T) -I)rf‖ p ～ K2r(f ,n-2r)w,p ～‖ w( Rn,(T)r f -f) ‖ p,where the weight function w (x) = ( 1 - x2 )- 12p, Rn,(T)r (f, x ) = n∑k=0 ( 1 - k2rn2r) αk (f) Tk ( x ), is the Riesz-type means of order r of Fourier-Chebyshev expansion off and K2r (f, tr) w,p is a new weighted K-functional defined byK2r(f,tr )w,p = inf g∈C2r-1,1](‖w(f-g)‖p+tr‖wP(D)rg‖p),Where the differential operator P( D ) = √1-x2 ddx √1-x2 d/dx.