Bernstein's comparison theorem and a problem of Braess
As a partial generalization of the Bernstein comparison theorem, \textit{D. Braess} [Aequationes Math. 12, 80-81 (1975; Zbl 0328.41012)] proved the theorem: Let \(f,g\in C^{n+1}[a,b]\) and assume that (1) \(0\leq g^{(n+1)}(x)\leq f^{n+1}(x),\) \(x\in [a,b]\). Then (2) \(\min_{s\in {\mathcal S}_{n,k}}\| g-s\|_{\infty}\leq \min_{s\in {\mathcal S}_{n,k}}\| f-s\|_{\infty}.\) The author extended this theorem to the case where (3) \(| g^{(n+1)}(x)| \leq f^{(n+1)}(x),\) \(x\in [a,b]\) replaces assumption (1). In this case (2) is no longer valid. He demonstrates this fact by determining the precise upper bound on the best approximation of the functions g satisfying (3), by splines of degree n with k variable knots. It is also shown that interpolation at \(n+k+1\) fixed, distinct points by splines of degree n with k fixed knots (these depend upon f) gives the same upper bound as that which is obtained for the best approximation from \({\mathcal S}_{n,k}\) to the class of functions g satisfying (3). The author also characterizes the best constant in the degree of approximation from \({\mathcal S}_{n,k}\), of functions in the Sobolev space \(W_{\infty}\!^{n+1}\) which satisfy \(| g^{(n+1)}|_{\infty}\leq 1.\)
scientific article; zbMATH DE number 3846148
scientific article