Geodesic
On integral Lebesgue constants of local splines with uniform knots
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 290-297
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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 $S$.
Mots-clés : Lebesgue constants
Keywords: local splines, boundary conditions. 
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     title = {On integral {Lebesgue} constants of local splines with uniform knots},
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V. T. Shevaldin. On integral Lebesgue constants of local splines with uniform knots. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 290-297. http://geodesic.mathdoc.fr/item/TIMM_2018_24_2_a26/

[1] Shevaldin V.T., Approksimatsiya lokalnymi splainami, Izd-vo UrO RAN, Ekaterinburg, 2014, 198 pp.

[2] Shevaldin V.T., Shevaldina O.Ya., “Konstanta Lebega lokalnykh kubicheskikh splainov s ravnootstoyaschimi uzlami”, Sib. zhurn. vychisl. matematiki, 20:4 (2017), 445–451 | DOI

[3] Subbotin Yu.N., Telyakovskii S.A., “Normy v $L$ periodicheskikh interpolyatsionnykh splainov s ravnootstoyaschimi uzlami”, Mat. zametki, 74:1 (2003), 108–117 | DOI | MR | Zbl

[4] Guin Shaohui, Liu Yongping, “Asymptotic estimate for the Lebesgue constant of cardinal $\cal L$-spline interpolation operator”, East J. of Approx., 13:3 (2007), 331–355 | MR

[5] Subbotin Yu.N., “Nasledovanie svoistv monotonnosti i vypuklosti pri lokalnoi approksimatsii”, Zhurn. vychisl. matematiki i mat. fiziki, 33:7 (1993), 996–1003 | MR | Zbl

[6] Kostousov K.V., Shevaldin V.T., “Approksimatsiya lokalnymi trigonometricheskimi splainami”, Mat. zametki, 77:3 (2005), 354–363 | DOI | MR | Zbl

[7] Kostousov K.V., Shevaldin V.T., “Approximation by local exponential splines”, Proc. Steklov Inst. Math., Suppl. 1, 2004, S147-S157 | MR | Zbl