Odd degree splines of higher order (Q1324765)

On the interval \([0,1]\), consider a function \(f(x)\) that is continuous to the order \(n+1\). An arbitrary partition of this interval is given as follows: \[ 0= x_ 0< x_ 1<\dots <x_ n=1. \] The problem is to determine a piecewise polynomial \(s(x)\) of degree \(2n+1\) on each subinterval \([x_ i, x_{i+1}]\), \(i=0,\dots, N-1\), being continuous to order \(n\) on the whole interval \([0,1]\), and stisfying three interpolating conditions. The theorem of existence and uniqueness of \(s(x)\) is proved and the theorem of error bounds of \(s^{(r)} (x)\) and \(f^{(r)} (x)\) \((r=0,\dots, n+1)\) is given.