Geodesic
On the approximation of $\exp(-x)$ on the half-axis by spline functions in three-point rational interpolants
Daghestan Electronic Mathematical Reports, Tome 11 (2019), pp. 32-37

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For the function $f(x)=\exp(-x)$, $x\in [0,+\infty)$ on grids of nodes $\Delta: 0=x_0$ with $x_n\to +\infty$ we construct rational spline-functions such that $R_k(x,f, \Delta)=R_i(x,f)A_{i,k}(x)+R_{i-1}(x, f)B_{i,k}(x)$ for $x\in[x_{i-1}, x_i]$ $(i=1,2,\dots)$ and $k=1,2,\dots$ Here $A_{i,k}(x)=(x-x_{i-1})^k/((x-x_{i-1})^k+(x_i-x)^k)$, $B_{i,k}(x)=1-A_{i,k}(x)$, $R_j(x,f)=\alpha_j+\beta_j(x-x_j)+\gamma_j/(x+1)$ $(j=1,2,\dots)$, $R_j(x_m,f)=f(x_m)$ при $m=j-1,j,j+1$; we take $R_0(x,f)\equiv R_1(x,f)$. Bounds for the convergence rate of $R_k(x,f, \Delta)$ with $f(x)=\exp(-x)$, $x\in [0,+\infty)$, are found.
Keywords: interpolation spline, rational spline, approximation on semi-axis. 
@article{DEMR_2019_11_a4,
     author = {A.-R. K. Ramazanov and V. G. Magomedova},
     title = {On the approximation of $\exp(-x)$ on the half-axis by spline functions in three-point rational interpolants},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {32--37},
     publisher = {mathdoc},
     volume = {11},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2019_11_a4/}
}
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A.-R. K. Ramazanov; V. G. Magomedova. On the approximation of $\exp(-x)$ on the half-axis by spline functions in three-point rational interpolants. Daghestan Electronic Mathematical Reports, Tome 11 (2019), pp. 32-37. http://geodesic.mathdoc.fr/item/DEMR_2019_11_a4/