Convergence bounds for splines for three-point rational interpolants of continuous and continuously differentiable functions

For functions $f(x)$ continuous on an interval $[a,b]$ and grids of pairwise different nodes $\Delta\colon a=x_0<x_1<\dots<x_N=b$ $(N\geqslant 2)$, we study the convergence rate of piecewise rational functions $R_{N,1} (x)=R_{N,1}(x,f)$ such that, for $x\in [x_{i-1}, x_i]$ ($i=1,2,\dots,N$), we have $R_{N,1}(x)=(R_i(x)(x-x_{i-1})+R_{i-1}(x)(x_i-x))/(x_i-x_{i-1})$, where $R_i(x)=\alpha_i+\beta_i(x-x_i)+\gamma_i/(x-g_i)$ ($i=1,2,\dots,N-1$); the coefficients $\alpha_i$, $\beta_i$, and $\gamma_i$ are defined by the conditions $R_i(x_j)=f(x_j)$ for $j=i-1,i,i+1$; and the poles $g_i$ are defined by the nodes. It is assumed that $R_0(x)\equiv R_1(x)$ and $R_N(x)\equiv R_{N-1} (x)$. Bounds for the convergence rate of $R_{N,1} (x,f)$ are found in terms of certain structural characteristics of the function:
(1) the third-order modulus of continuity in the case of uniform grids;
(2) the variation and the modulus of change of the first and second derivatives in the case of continuously differentiable functions $f(x)$; here, the bounds in terms of the variation have the order of the best polynomial spline approximations.