# 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.

REFERENCES

1.   Shevaldin V.T. Approximaciya lokalnymi splajnami [Local approximation by splines]. Ekaterinburg: UrO RAN Publ., 2014, 198 p. (in Russian).

2.   Shevaldin V.T., Shevaldina O.Ya. The Lebesgue constant of local cubic splines with equally-spaced knots. Siberian J. Num. Math., 2017, vol. 20, no. 4, pp. 445–451. doi: 10.15372/SJNM2017040 .

3.   Subbotin Yu.N., Telyakovskii S.A. Norms on $L$ periodic interpolation splines with equidistant nodes. Math. Notes., 2003, vol. 74, no. 1, pp. 100–109.

4.   Guin Shaohui, Liu Yongping. Asymptotic estimate for the Lebesgue constant of cardinal $\cal L$-spline interpolation operator. East J. of Approx., 2007, vol. 13, no. 3, pp. 331–355.

5.   Subbotin Yu.N. Inheritance of monotonicity and convexity in local approximation. Comput. Math. Math. Physics, 1993, vol. 33, no. 7, pp. 879–884.

6.   Kostousov K.V., Shevaldin V.T. Approximation by local trigonometric splines. Math. Notes, 2005, vol. 77, no. 3, pp. 326–334.

7.   Kostousov K.V., Shevaldin V.T. Approximation by local exponential splines. Proc. Steklov Inst. Math., 2004, Suppl. 1, pp. S147–S157.