三角插值中的线性求和问题

On a linear summation problem in trigonometric interpolation

本文通过选取求和因子构造出和式型三角插值多项式Ｈｎ（ｆ，ｒ，ｘ）（ｒ为奇自然数），使其在全实轴上一致地收敛到以２π为周期的连续函数ｆ（ｘ），且Ｈｎ（ｆ，ｒ，ｘ）对Ｃｎ２π（ｌ≤ｒ）连续函数类的逼近均达到最佳收敛阶．Ｈｎ（ｆ，ｒ，ｘ）的饱和阶为１／ｎ（ｒ＋１），饱和函数类为ｆ（ｒ）（ｘ）∈Ｌｉｐｍｌ．

In this paper the trigonometric interpolation summation polynomials Hn(f,r,x) (r is an odd natural number) are constructed. If the function f(x)∈C2n, then Hn(f,r,x) converge the function f(x) uniformly on (-∞,∞), and the convergence order is the best if f(x)∈C2n(l≤r); the saturable order of H. (f,r,x) is and the saturated function class is f(r)(x)∈LipM1.