Splines for three-point rational interpolants with autonomous poles

For arbitrary grids of nodes $\Delta: a=x_0<x_1<\dots<x_N=b$ $(N\geqslant 2)$ smooth splines for three–point rational interpolants are constructed, the poles of interpolants depend on nodes and the free parameter $\lambda$. Sequences of such splines and their derivatives for all functions $f(x)$ respectively of the classes of $C_{[a,b]}^{(i)}$ $(i=0,1,2)$ under the condition $\|\Delta\| \to 0$ uniformly in $[a,b]$ converge respectively to $f^{(i)}(x)$ $(i=0,1,2)$ (depending on the parameter $\lambda$). Bonds for the convergence rate are found in terms of the distance between the nodes.