Optimal error bounds for Hermite interpolation (Q1098025)

Let \(u(x)\in C^ 3[0,1]\) and \(v_ 3(x)\) the unique Hermitian interpolation polynomial of degree \(\leq 3\) satisfying \(v_ 3(0)=u(0)\), \(v_ 3(1)=u(1)\), \(v_ 3'(0)=u'(0)\), \(v_ 3'(1)=u'(1)\). The authors obtain pointwise bounds for the error \(e(x)=v_ 3(x)-u(x)\) and its derivatives in terms of \(L=\max (| u\prime''(x)|;\quad 0\leq x\leq 1)\) or in terms of \(M=\max (| u''(x)|\); \(0\leq x\leq 1)\). These estimates complete those obtained by \textit{P. G. Ciarlet}, \textit{M. H. Schultz}, \textit{R. S. Varga} [Numer. Math. 9, 394-430 (1967; Zbl 0155.204)] and \textit{G. Birkhoff}, \textit{A. Priver} [J. Math. Phys. 46, 440-447 (1967; Zbl 0176.142)]. In addition, these estimates are the best possible.