关于用多维样条逼近的饱和度

<正> 作者在工作1]中,研究了一维样条逼近的饱和度问题,得到了下述定理: 定理A 设{△_k}是区间a,b]的一分划序列,‖△_k‖→0(k→∞),R_(△k)≤β<∞(k=1,2,…),若f(x)∈c~na,b],S_(△_k)(x)是(n-1)次多项式样条,如果 对任一p成立,(p=0,1,…,n)则 D~nf(x)≡0. 本文是1]的续篇,对多维样条进行研究,可得一些类似的结果,为明确起见,我们仅对二维情形进行讨论.

ON THE SATURATION OF MULTIDIMENSIONAL SPLINE APPROXIMATION

In this paper we consider the problem of saturation by two-dimensional spline approximation associated with {△~((k))}, where {△~((k))} is a sequence of partition of Ω=a, b] × c, d].The main results we have obtained are:Theorem 1. Let f(x, y)∈C~((m,n)) (Ω), S_(△(k)) be the first Type of degree (m-1) × (n - 1) of blended Spline, R_(△(k)) ≤ β < ∞, (k = 1, 2,…) and ‖△_x~((k))‖)→0, ‖△_y~((k))‖→0(k→∞).If max holds for some(p,l),(0≤p≤m,0≤l≤n),thenTheorem 2. Let f(x,y)be Comtinuous On and S_(△(k) be the Second Type of degree(m-1)×(n-1)of spline andif max then