Quadratic spline interpolation (Q5899858)

Let P be a mesh of [0,1] given by \(0=x_ 0<x_ 1<...<x_ n=1\), \((x_ j-x_{j-1}=h\), \(j=1,...,n)\) and let \(S(2,P)=\{s:\) \(s_ i\in \pi_ 2\), \(s\in C^ 1[0,1]\), \(s^{(j)}(0)=s^{(j)}(1)\), \(j=0,1\}\) be the class of all 1-periodic quadratic splines, corresponding to the mesh P. By a result of \textit{A. Sharma} and \textit{J. Tzimbalario} [J. Approximation Theory 19, 186-193 (1977; Zbl 0354.41006)] it follows that if f is a 1- periodic locally integrable function, then there exists a unique spline function \(s\in S(2,P)\) such that \(s(u_ i)=f(u_ i)\), where \(u_ i=(x_{i-1}+x_ i)/2\), \(i=1,2,...,n\). The values \(m_ i=s'(x_ i)\) can be determined from a system which can be written in the matrix form as \(AM=F\), \(M=[m_ 1,...,m_ n]^ t\), \(F=[F_ 1,...,F_ n]^ t,\) \(F_ i=4[f(u_{i+1})-f(u_ i)]/h.\) The author gives the following delimitation for the norm of the inverse of \(A:\;\| A^{-1}\| \leq (1+r)(1-r)^{-1}(3+r)^{-1},\) \(r=\sqrt{8}-3\). If f''' is continuous non-negative and monotone, then \[ s(x)-f(x)=f\prime''(x)(x_ i+x_{i- 1}-2x)\cdot [(x_ i+x_{i-1}-2x)^ 2-h^ 2]/48+O(h^ 3), \] for every \(x\in (0,1)\).