A.-R.K. Ramazanov, A.K. Ramazanov, V.G. Magomedova. On the Gibbs phenomenon for rational spline functions ... P. 238-251

In the case of functions $f(x)$ continuous on a given closed interval $[a,b]$ except for jump discontinuity points, the Gibbs phenomenon is studied for rational spline functions $R_{N,1}(x)=R_{N,1}(x,f,\Delta, g)$ defined for a knot grid $\Delta: a=x_0<x_1<\dots<x_N=b$ and a family of poles $g_i\not \in [x_{i-1},x_{i+1}]$ $(i=1,2,\dots,N-1)$ by the equalities $R_{N,1}(x)= [R_i(x)(x-x_{i-1})+R_{i-1}(x)(x_i-x)]/(x_i-x_{i-1})$ for $x\in[x_{i-1}, x_i]$ $(i=1,2,\dots,N)$. Here the rational functions $R_i(x)=\alpha_i+\beta_i(x-x_i)+\gamma_i/(x-g_i)$ $(i=1,2,\dots,N-1)$ are uniquely defined by the conditions $R_i(x_j)=f(x_j)$ $(j=i-1,i,i+1)$; we assume that $R_0(x)\equiv R_1(x)$, $R_N(x)\equiv R_{N-1}(x)$. Conditions on the knot grid~$\Delta$ are found under which the Gibbs phenomenon occurs or does not occur in a neighborhood of a discontinuity point.

Keywords: interpolation spline, rational spline, Gibbs phenomenon

Received December 10, 2019

Revised  May  18, 2020

Accepted  May 25, 2020

A.-R.K. 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

A.K. Ramazanov, Cand. Sci. (Phys.-Math.), Bauman Moscow State Technical University (Kaluga Branch), Kaluga, 248000, Russia, e-mail: akramazanov@mail.ru

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

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 Cite this article as: A.-R.K. Ramazanov, A.K. Ramazanov, V.G. Magomedova. On the Gibbs phenomenon for rational spline functions. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 238-251.