On the Gibbs phenomenon for rational spline functions

A.-R. K. Ramazanov ; A.-K. K. Ramazanov ; V. G. Magomedova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 238-251

Résumé

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 $\Delta: a=x_0$ 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.

Détail

Mots-clés : interpolation spline
Keywords: rational spline, Gibbs phenomenon.