(0,2,3) and (0,1,3) interpolation by six degree splines
Denote by \(S^{(3)}_{m,6}\) the set of all spline functions \(S_{\Delta}\in C^ 3(I)\), \(I=[0,1]\), \(\Delta\) : \(0=x_ 0<...<x_ m=1\), a division of the interval I, such that \(S_{\Delta}\) is a polynomial of degree at most 6 on each interval \([x_{k-1},x_ k]\), \(k=1,2,...,m\). The authors consider two lacunary interpolation problems: Problem (0,2,3) - Find a spline function \(S_{\Delta}\in S^{(3)}_{m,6}\) which interpolates the values of a given function f on I and the values of its derivatives of order 2 and 3, at the knots of \(\Delta\). The problem (0,1,3) is defined similarly. The uniqueness and existence of the solutions of these problems are proved. Estimates of the error of approximation in terms of the modulus of continuity of the sixth derivative \(f^{(6)}\) of f are also given.
scientific article; zbMATH DE number 4034280