V. T. Shevaldin. Uniform Lebesgue constants of local spline approximation ... P. 292-299.

Let a function $ \varphi \in C^1[-h,h]$ be such that $ \varphi (0)= \varphi '(0)=0$, $ \varphi (-x)= \varphi (x)$ for $x\in [0;h])$, and $ \varphi (x)$ is nondecreasing on $[0;h]$. For any function $f: \mathbb R\to \mathbb R$, we consider local splines of the form $$S(x)=S_{ \varphi }(f,x)=\sum_{j\in \mathbb Z} y_j B_{ \varphi }\Big( x+\frac{3h}{2}-jh\Big)\quad (x\in \mathbb R),$$ where $y_j=f(jh)$, $m(h)>0$, and $$B_{ \varphi }(x)=m(h)\left\{\begin{array}{cl} \varphi (x),& x\in [0;h],\\[1ex] 2 \varphi (h)- \varphi (x-h)- \varphi (2h-x),& x\in [h;2h],\\ \varphi (3h-x),& x\in [2h;3h],\\[1ex] 0, & x\not\in [0;3h]. \end{array} \right.$$ These splines become parabolic, exponential, trigonometric, etc., under the corresponding choice of the function $ \varphi $. We study the uniform Lebesgue constants $L_{ \varphi }=\|S\|_C^C$ (the norms of linear operators from $C$ to $C$) of these splines as functions depending on $ \varphi $ and $h$. In some cases, the constants are calculated exactly on the axis $\mathbb R$ and on a closed interval of the real line (under a certain choice of boundary conditions from the spline $S_{ \varphi }(f,x)$). Keywords: Lebesgue constants, local splines, three-point system.

The paper was received by the Editorial Office on June 2, 2017.

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


1.   Ahlberg J., Nilson E.N., Walsh J. The theory of Splines and Their Applications. New York, London: Academic Press, 1967, 284 p. ISBN: 0120447509 . Translated under the title Teoriya splainov i ee prilozheniya, Moscow, Mir Publ., 1972, 318 p.

2.   Shevaldin V.T. Lower bounds on width of classes of sourcewise representable functions. Proc. Steklov Inst. Math., 1990, vol. 189, pp. 217–234.

3.   Rvachev V.A. Compactly supported solutions of functional-differential equations and their applications. Russian Math. Surveys, 1990, vol. 45, no. 1, pp. 87–120. doi: 10.1070/RM1990v045n01ABEH002324 .

4.   Leont’ev V.L. Ortogonal’naya finitnye funkzii i chislennye metody [Orthogonal compactly supported functions and numerical methods]. Ul’yanovsk: Ul’yanovskij Gosudarstvennyj Universitet Publ., 2003, 177 p. ISBN: 5-88866-144-9/hbk .

5.   Kvasov B.I. Metody isogeometricheskoi approximacii [Methods for isogeometric approximation]. Moscow: Fizmatlit Publ., 2006, 360 p. ISBN: 5-9221-0733-Х .

6.   Dem’yanovich Y.K. Wavelets basic of Bφ-splines for erregular net. Mat. Mod., 2006, vol. 18, no. 10, pp. 123–126 (in Russian).

7.   Shevaldin V.T. Approksimatsiya lokal’nymi splainami [Local approximation by splines]. Ekaterinburg: UrO RAN Publ., 2014, 198 p.

8.   Zav’yalov Yu.S., Kvasov B.I., Miroshnichenko V.L. Metody splain-funktsii [Methods of spline functions]. Moscow, Nauka Publ., 1980, 352 p.

9.   Subbotin Yu.N. Inheritance of monotonicity and convexity in local approximations. Comput. Math. Math. Phys., 1993, vol. 33, no. 7, pp. 879–884.

10.   Shevaldin V.T. Approximation by local parabolic splines with arbitrary knots. Sib. Zh. Vychisl. Mat., 2005, vol. 8, no. 1, pp. 77–88.

11.   Kostousov K.V., Shevaldin V.T. Approximation by local exponential splines. Proc. Steklov Inst. Math., 2004, Suppl. 1, pp. 147–157.

12.   Kostousov K.V., Shevaldin V.T. Approximation by local trigonometric splines. Math. Notes, 2005, vol. 77, no. 3, pp. 326–334. doi: 10.1007/s11006-005-0033-z .