On \(L_ p\)-boundedness of the \(L_ 2\)-projector onto splines
Let \(\Delta_ n= \{a= t_ 0< t_ 1<\cdots< t_ n= b\}\) be a partition of the interval \([a,b]\), \(S_{m-1}(\Delta_ n)\) be the space of splines of degree \(m-1\) with deficiency 1 on the mesh \(\Delta_ n\), \(P_ S\) be an orthogonal projection onto \(S_{m-1}(\Delta_ n)\) defined by \[ \int^ b_ a [f(t)- P_ S(f,t)] \sigma(t) dt\quad \forall\sigma\in S_{m-1}(\Delta_ n). \] It is proved that the \(L_ p\)-norm of the operator \(P_ S\) is uniformly bounded for any mesh \(\Delta_ n\) at least for \(p\) close to 2 and that it is uniformly bounded for all \(p\in [1,\infty]\) in meshes \(\Delta_ n\) with a fixed number of nodes \(n\).
scientific article; zbMATH DE number 636074