The method of integral relations for an estimation of an approximation error in some Sobolev spaces by interpolating spline-functions (Q1093111)

Let \(f\in W_{\infty}^{k-1}<a,b>\), \(-\infty <a<b<\infty\), \(S(2n-1,2n- r,\Delta) \ni S\) be an interpolating polynomial spline-function satisfying some generalized type boundary conditions. The author presents the estimates of the deviation \(D^{\alpha}(f-s)\) in terms of integral norms and some moduli of continuity connected with the partition \(\Delta\) of the interval \(<a,b>\). Presented estimates generalize the earlier results by \textit{R. S. Varga} and \textit{M. H. Schultz} [Numer. Math. 10, 345-369 (1967; Zbl 0183.444)], \textit{B. K. Swartz} and \textit{R. S. Varga} [J. Approximation Theory 6, 6-49 (1972; Zbl 0242.41008)] and the author [Mat. Zametki 22, 593-600 (1977; Zbl 0374.41004)].