Error bounds for two even degree tridiagonal splines (Q808351)

\textit{M. J. Marsden} [Bull. Am. Math. Soc. 80, 903-906 (1974; Zbl 0295.41005)] has investigated interpolation of continuous, periodic functions f by quadratic periodic splines on a finite interval [a,b]. The error bounds he derived are improved in this paper. In addition interpolation by quadratic periodic splines is considered. These are fourth order polynomials S on each subinterval \((x_{i-1},x_ i)\), \(i=1,...,k (x_ 0=a\), \(x_ k=b)\) with \(a<x_ 1<...<x_{k-1}<b\) such that \(S\in C^ 2[a,b],\) \(S''(x_ i)=0\) and \(S(z_ i)=f(z_ i)\) for \(i=1,...,k\) where \(z_ i={1\over 2} (x_{i-1}+x_ i),\) and \(S(a)=S(b), S'(a)=S'(b).\) As an improvement of corresponding results by Marsden it is proved as Theorem \(1'\) that \(| S(x_ i)| \leq (8/5)| f(z_ i)|,| f(x_ i)-S(x_ i)| \leq (8/5)w(f,(h/2))\) for \(i=1,...,k\), and \(\max_{x\in [a,b]}| f(x)- S(x)| \leq (13/8)w(f,(h/2))\) where \(h=\max \{| x_ i-x_{i- 1}| \quad i=1,...,k\}\) and w(f,d) denotes the modulus of continuity of f. In Theorem \(3'\) error estimates are given for the interpolation of twice continuously differentiable functions by quadratic splines which also improve corresponding results by Marsden for quadratic splines.