# A.-R.K. Ramazanov, V.G. Magomedova. Convergence bounds for splines for three-point rational interpolants of continuous and continuously differentiable functions ... С. 224-233.

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 pol

Keywords: splines, interpolation splines, rational splines.

The paper was received by the Editorial Office on April 17, 2017.

Abdul-Rashid Kehrimanovich Ramazanov, Dr. Phys.-Math., Prof., Dagestan State University, the Republic of Dagestan, Makhachkala, 367002 Russia; Dagestan Scientific Center RAN, the Republic of Dagestan, Makhachkala, 367025 Russia, e-mail: ar-ramazanov@rambler.ru

Vazipat Gusenovna Magomedova, Cand. Sci. (Phys.-Math.), Dagestan State University, the Republic of Dagestan, Makhachkala, 367002 Russia, e–mail: vazipat@rambler.ru

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