Splines on rational interpolants

For a function continuous on a given interval (or periodic) we construct $n$-point ($n=2,3,4$) rational interpolants and rational splines by means of of these interpolants. The sequences of the splines by the n-point interpolants for $n = 2$ and $n=3$ converges uniformly on the entire interval to the function itself for any sequence of grids with a diameter tending to zero. For $n= 3$ this property of unconditional convergence is also transmitted to the first derivatives, and for $n = 4$ – to the first and second derivatives.
We also give estimates of the convergence rate.