Fourier—Chebyshev算子及其Norlund平均

FOURIER—CHEBYSHEV OPERATOR AND ITS N ORLUND MEAN

讨论了 Fourier—Chehyshev算子及其Norlund平均对有界变差函数的逼近.

The approximation of function of bounded variation by Fourier-Chebyshev operator and its Norlund mean are discussed. The main result is as following:Theorem. If f(x)∈B.V.[-1,1] and S_n(f;x)=integral from n=-1 to 1 (f(y)/(1-y^2)^(1/2)K_n(x,y))dy is the Fourier-Chebyshev operator, then where K_n(x,y)=sum form n=0 to n(T_k(x)T_k(y), T_n-Chebyshev polynomialf(y)-f(x-0), -1≤y<x,and A_x(y)={0, y=x,f(y)-f(x+0), x<y≤1.Corollary. If f(x)∈B. V. [-1,1] and p_n～1/n~α, (?)(A_x)=O(|δ|~α) 0<α<1,then for the Norlund mean: N_n(f;x)=1/P_n sum form n=0 to n p_n-r^S_i(f;x)we have N_n(f;x)-1/2(f(x+0)+f(x-0))| ≤44/π(1-x^2)·1/P_n sum from i=2 to n(p_(n-i))·1/i sum form n=1 to i ■(A_x)+O(Inn/n)=O(1/n~α),-1<x<1, n>1.