【摘要】 <正> 设给定函数F（t）,它在[0,1]上有各阶导数,作级数: sum from j=0 to ∞[F^2j（0）f_2j+1（t）+F^2j（1）g_2j+1（t）],0≤t≤1, （1）其中f_2j+1（t）与g_2j+1（t）为[1]中定义的2n+1次多项式。[2]中给出了下述定理: 定理A.已给函数F（t）,0≤t≤1。若偶阶导数序列{F^2j（t）}在[0,1]上一致有界,即存在M>0,使得更多还原

【Abstract】 In the paper "Uniform Convergence of the sequences of Functions {f_2j+1（t）} and{g_2i+1（t）}", the author has obtained a theorem: The sequence P_2n+1（x）=sub from j=0 to n[F^2j（0）f_2j+1（x）+F^2j（1）g_2j+1（x）] converges uniformly to F（x） in [0,1], if the sequence of even orderderivatives of F（x） in [0, 1] is bounded uniformly. In this paper, we shall improve the abovetheorem. We shall give the following results: （1） F（x）=P_2n+1（x）+F²ⁿ⁺²（ξ）/（2n+2）!φ_2n+2（x）,where φ_2n+2（x） is a polynomial of degree （2n+2）. （2） P_2n+1（x） and its derivatives convergeuniformly to F（x） and its correlative derivatives in [0, 1] respectively, if F^2j（x）=o(2^2j).更多还原