A generalization of Jackson's inequality (Q792553)

In this article the following assertions are proved: Theorem 1. Let \(S_{k,n}\) be the space of smooth splines of degree k-1 (i.e., \(S_{k,n}\subset C^{k-2})\) with n equally spaced knots on the interval [0,1]. Then \((1)\quad \inf_{s\in S_{k,n}}\| f-s\|_ p\leq |(C\cdot 1/(n+k))\| f'\|_ p,\) where C is an absolute constant less than 10. Theorem 2. Let \(S_{k,t}\) be the space of smooth splines on a mesh \(t=\{0=t_ 0<t_ 1<...<t_ n<t_{n+1}=1\}\). Then \(\inf_{s\in S_{k,t}}\| f-s\|_ p\leq C'(({\bar \Delta})^{- 1}+k)^{-j}\| f^{(j)}\|_ p,\) \(k\geq j\), where C' only depends on j, and \({\bar \Delta}=\max_{0\leq i\leq n}(t_{i+1}-t_ i).\) In these theorems \(f^{(j)}\) denotes the derivative of order j of the function f, 1\(\leq p\leq \infty\), and \(\| \cdot \|\) denotes the usual norm of the space \(L_ p[0,1]\). For \(n=0\), (1) reduces to Jackson's well-known inequality for polynomial approximation.