On integral Lebesgue constants of local splines with uniform knots

V. T. Shevaldin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 290-297

Résumé

We study the stability properties of generalized local splines of the form $$ S(x)=S(f,x)=\sum_{j\in \mathbb Z} y_j B_{\varphi}\Big( x+\frac{3h}{2}-jh\Big)\quad (x\in \mathbb R), $$ where $\varphi\in C^1[-h,h]$ for $h>0$, $\varphi(0)=\varphi'(0)=0$, $\varphi(-x)=\varphi(x)$ for $x\in [0;h]$, $\varphi(x)$ is nondecreasing on $[0;h]$, $f$ is an arbitrary function from $\mathbb R$ to $\mathbb R$, $y_j=f(jh)$ for $j\in \mathbb Z$, and $$ B_{\varphi}(x)=m(h)\left\{\begin{array}{cl} \varphi(x), {\} x\in [0;h],\\[1ex] 2\varphi(h)-\varphi(x-h)-\varphi(2h-x), {\} x\in [h;2h],\\[1ex] \varphi(3h-x), {\} x\in [2h;3h],\\[1ex] 0, {\} x\not\in [0;3h]\end{array}\right. $$ with $m(h)>0$. Such splines were constructed by the author earlier. In the present paper we calculate the exact values of their integral Lebesgue constants (the norms of linear operators from $l$ to $L$) on the axis $\mathbb R$ and on any segment of the axis for a certain choice of the boundary conditions and the normalizing factor $m(h)$ of the spline $S$.

Détail

Mots-clés : Lebesgue constants
Keywords: local splines, boundary conditions.