(0,2,3) and (0,1,3) interpolation through splines (Q1105796)

Let \(\Delta:0=x_ 0<x_ 1<...<x_{m-1}<x_ m=1\) be the uniform partition of the interval \(I=[0,1]\) with \(x_ k=k/m\). The authors consider the class \(S_{m,6}^{*(3)}\) of splines \(S_{\Delta}\) which satisfy two conditions; (i) \(S_{\Delta}\in C^ 3(I)\) and (ii) in each interval \([x_ k,x_{k+1}]\), \(S_{\Delta}\in \Pi_ 6\) except in one of the end intervals, say \([x_ 0,x_ 1]\) where \(S_{\Delta}\in \Pi_ 7\). They first prove the existence and uniqueness of splines \(S_{\Delta}\) and \(\tilde S_{\Delta}\in S_{m,6}^{*(3)}\) which satisfy (0,2,3) and (0,1,3) interpolation conditions respectively. They also obtain bounds for the errors \(| S_{\Delta}^{(q)}- f^{(q)}|\) and \(| \tilde S_{\Delta}^{(q)}-f^{(q)}|\), \(q=0,1,...,5\) in the case when \(f\in C^{(6)}(I)\). It would be interesting to compare these with the earlier results of \textit{J. Györvari} [ibid. 42, 25-53 (1983; Zbl 0529.41005)].