A Convergent Family of Bivariate Floater-Hormann Rational Interpolants

On the approximation of $ \ exp (-x) $ on the half-axis by spline functions in three-point rational interpolants

Daghestan Electronic Mathematical Reports ◽

10.31029/demr.11.4 ◽

2019 ◽

pp. 32-37

Author(s):

Abdul-Rashid Ramazanov ◽

V.G. Magomedova

Keyword(s):

Convergence Rate ◽

Spline Functions ◽

Rational Spline ◽

Rational Interpolants ◽

Half Axis

For the function $f(x)=\exp(-x)$, $x\in [0,+\infty)$ on grids of nodes $\Delta: 0=x_0<x_1<\dots $ with $x_n\to +\infty$ we construct rational spline-functions such that $R_k(x,f, \Delta)=R_i(x,f)A_{i,k}(x)\linebreak+R_{i-1}(x, f)B_{i,k}(x)$ 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.

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Splines on rational interpolants

Daghestan Electronic Mathematical Reports ◽

10.31029/demr.4.3 ◽

2015 ◽

pp. 21-30

Author(s):

Abdul-Rashid Ramazanov ◽

Vazipat Magomedova ◽

◽

Keyword(s):

Rational Interpolants

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Perturbation of nodes and poles in certain rational interpolants

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700026459 ◽

1987 ◽

Vol 36 (2) ◽

pp. 177-185

Author(s):

M.A. Bokhari

Keyword(s):

Rational Interpolants

Recently, E.B. Saff and A. Sharma proved a result on Walsh's type equiconvergence for certain sequences of rational interpolants having uniformly distributed poles and nodes of interpolation. Here we examine the sensitivity of this result to a slight perturbation of the poles and nodes. Some problems related to our work are also formulated.

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