r 阶导数为ΦBV中函数的Fourier级数收敛速度估计

AN ESTIMATE FOR FOURIER SERIES OF FUNCTION WITH R-DERIVATIVES OF φ-BOUNDED VARIATION

对于周期为2π并且 r 阶导数为φ-有界变差函数,我们证明了:|Sn(f,x)-f(x)-(sin(1/2)π/rπn^r)(f_R^(r)(x)-f_L^(r)(x))|≤(3/n^(r+1)sum from n to k=1 V_φ( _x,[0,π/k]))+(2|sin(r/2)π|/πn^(r+1))|f_R^(r)(x)-(f_L^(r)(x)|,其中

For 2π-periodic functions whose r-th derivatives are of Φ-Bounded Variation we prove that: |S_n(f,x)-f(x)-(sinr/2π)/(rπn^r)|(f_R^((r))(x)-f_L((r))-f_L((r))(x))|≤3/(n^(r+1))sum from k=1 to n V_Φ(φ_x[O,π/K])+(2|sin(rπ)/2)|/(πn^(r+1))|f_R^((r))(x)-f_L^((r))(x|f∈ΦBV∩V_r.