In this paper we derive stability estimates in $L^{2}$- and $L^{\infty}$- based Sobolev spaces for the $L^{2}$ projection and a family of quasiinterolants in the space of smooth, 1-periodic, polynomial splines defined on a uniform mesh in $[0,1]$. As a result of the assumed periodicity and the uniform mesh, cyclic matrix techniques and suitable decay estimates of the elements of the inverse of a Gram matrix associated with the standard basis of the space of splines, are used to establish the stability results.

翻译：在本文中,我们得出以美元为基值的Sobolev空间的稳定性估计值,即2美元预测值和以美元为基值的Sobolev空间的稳定性估计值,以及在单一网格($10,1美元)上定义的光滑、周期性、多环样条形的准中间体群,由于假定的周期和统一的网格、循环矩阵技术和与样板空间标准基数相关的格列矩阵各要素的适当的衰减估计值,因此,确定了稳定性结果。