On the stability of the $L^{2}$ projection and the quasiinterpolant in the space of smooth periodic splines

TL;DR

This paper establishes stability estimates for the $L^{2}$ projection and quasiinterpolants in Sobolev spaces for smooth periodic splines, using cyclic matrix techniques and decay estimates of Gram matrix inverses.

Contribution

It provides new stability bounds for projections and quasiinterpolants in periodic spline spaces, leveraging the structure of uniform meshes and cyclic matrix analysis.

Findings

Stability estimates in $L^{2}$ and $L^{}$ Sobolev spaces.

Use of cyclic matrix techniques for stability analysis.

Decay estimates of Gram matrix inverse elements.

Abstract

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.