Hermite-Fejer插值算子的平均收敛性

Mean Convergence Properties for Hermite-Fejér Interpolating Operators

讨论ＨｅｒｍｉｔｅＦｅｊéｒ插值算子Ｈ２ｎ－１（ｆ，ｘ）在Ｌｐ空间上平均收敛性，得到平均收敛的几个充要条件．其中之一：Ｈ２ｎ－１（ｆ（ｘ），ｘ）平均收敛于ｆ（ｘ）的充分必要条件是：‖Ｈ２ｎ－１‖ｐ有界，并且ｌｉｍｎ→∞‖∑ｎｋ＝１Ｈ２ｎ－１（ｘｉ，ｘ）－ｘｉ‖ｐ＝０，（ｉ＝１，２）．

L pmean convergence of Hermite-Fejér operators is investigated.Sufficient and necessary conditions are given.One of them is: H 2n-1 (w,f,x) mean convergence to f(x) if and only if the norms of ‖H 2n-1 ‖ p are uniformly bounded for all n and lim n→∞‖∑nk=1H 2n-1 (x i,x)-x i‖ p=0,(i=1,2).