Local behaviour of the derivative of a mid point cubic spline interpolator (Q1105139)

Let \(\Delta:0=x_ 0<x_ 1<...<x_ n=1\) be an equidistant partition of the interval \([0,1]\) and let \(p=x_ i-x_{i-1}\). Denote by \(P_ 3\) the class of all real-valued functions defined on \([0,1]\) whose restrictions to \([x_{i-1},x_ i]\) are polynomials of degree \(\leq 3\), for \(i=1,2,...,n\). Put \(S(3,\Delta)=\{s\in P_ 3,s\in C^ 2[0,1],s^{(j)}(0)=s^{(j)}(1)\), \(j=0,1,2\}\) and \(S^*(3,\Delta)=\{s\in S(3,\Delta)\), \(s''(0)=0\}\). The authors prove that if \(f\in C\) 4[0,1] is a periodic function of period 1 with \(f^{(4)}\) a non-negative and monotonic function and \(s\in S^*(3,\Delta)\) is such that \(s(t_ i)=f(t_ i)\), \(t_ i=(x_ i+x_{i-1})/2\), \(i=1,2,...,n\), then for every \(x\in (0,1)\) the following asymptotic evaluation: \[ s'(x)-f'(x)=f^{(4)}(x)[(t_{i+1}-x)^ 4- (t_ i-x)^ 4)/p-p((x_{i+1}-x)^ 2+ \] \[ +10.08(x_ i-x)^ 2+12.92(x-x_{i-1})^ 2-13.92p^ 2)/4]/24+\sigma (p^ 3), \] holds for \(n\to \infty\).