V.T. Shevaldin. On integral Lebesgue constants of local splines with uniform knots ... P. 290-297

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 $S$.

Keywords: Lebesgue constants, local splines, boundary conditions.

The paper was received by the Editorial Office on February 15, 2018.

Funding Agency:

Russian Science Foundation (Grant Number 14-11-00702).

Valerii Trifonovich Shevaldin, Dr.Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, e-mail: Valerii.Shevaldin@imm.uran.ru.


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