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.

中文翻译：

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光滑周期样条空间中$L^{2}$投影和拟插值的稳定性

在本文中，我们在基于 $L^{2}$- 和 $L^{\infty}$- 的 Sobolev 空间中推导了 $L^{2}$ 投影的稳定性估计以及平滑空间中的准互溶族，在 $[0,1]$ 中的均匀网格上定义的 1-周期多项式样条。由于假定的周期性和均匀网格，循环矩阵技术和与样条空间的标准基相关联的 Gram 矩阵的逆元素的适当衰减估计被用于建立稳定性结果。