Error of the approximation by interpolation polynomials of small degrees on n-simplices
Let \(S_{h,n}\) be the set of all n-simplexes \(\Delta_ n\) in \({\mathbb{R}}^ n\) with diam \(\Delta\) \({}_ n\leq h\). If \(\Delta_ n\in S_{h,n}\), let \(C_ M^{k+1}(\Delta_ n)\) be the set of all \(C^ k\) functions on \(\Delta_ n\) with \(\| f^{(k+1)}(x)\| \leq M\) for \(x\in \Delta_ n\), and let \(p_ k(f)\) be the polynomial of degree k in n variables that interpolates \(f\in C_ M^{k+1}(\Delta_ n)\) in vertices of \(\Delta_ n\), if \(k=1\), or else in vertices of \(\Delta_ n\) and the middle points of its sides, if \(k=2\). Set \[ \alpha_{k,n}:=\sup \{\sup | f-p_ k(f)| (\Delta_ n)M^{-1}h^{-k-1}:f\in C_ M^{k+1},\Delta_ n\in S_{h,n},h>0\}. \] The author proves that 1) for each n, \(\alpha_{1,n}=n/4(n+1);\) 2) if \(n\geq 2\), then \(n\sqrt{2n}/48(n+1)\sqrt{n+1}\leq \alpha_{2,n}\leq 5n/16(n+1).\)
scientific article; zbMATH DE number 4214643