Geodesic
Approximation by local exponential splines with arbitrary nodes
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 9 (2006) no. 4, pp. 391-402.

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For the class of functions $W_{\infty}^{\mathcal L_2}[a,b]=\{f\colon f'\in AC,\quad\|\mathcal L_2(\mathcal D)f\|_{\infty}\leq 1\}\quad(\mathcal L_2(\mathcal D)=\mathcal D^2-\beta^2 I,\beta>0$, $\mathcal D$ is operator of differentiation) a new noninterpolating linear method of local exponential spline-approximation with arbitrary nodes is constructed. This method has some smoothing properties and inherits monotonicity and generalized convexity of the data (values of a function $f\in W_{\infty}^{\mathcal L_2}$ at the grid points). The error of approximation in a uniform metric of a class of functions $W_{\infty}^{\mathcal L_2}$ by these splines is exactly determined. 
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E. V. Shevaldina. Approximation by local exponential splines with arbitrary nodes. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 9 (2006) no. 4, pp. 391-402. http://geodesic.mathdoc.fr/item/SJVM_2006_9_4_a6/

[1] Piegl L., Tiller W., The NURBS Book, Springer, Berlin etc., 1997 | Zbl

[2] Lyche T., Schumaker L. L., “Local spline approximation methods”, J. Approxim. Theory, 15:4 (1975), 294–325 | DOI | MR | Zbl

[3] Zavyalov Yu. S., Kvasov B. I., Miroshnichenko V. L., Metody splain-funktsii, Nauka, M., 1980 | MR

[4] Kvasov B. I., Izogeometricheskaya approksimatsiya splainami. Uchebnoe posobie, Novosibirskii gosuniversitet, Novosibirsk, 1998

[5] Miroshnichenko V. L., “Dostatochnye usloviya monotonnosti i vypuklosti dlya interpolyatsionnykh parabolicheskikh splainov”, Vychislitelnye sistemy, 142, 1991, 3–14 | MR

[6] Subbotin Yu. N., “Nasledovanie svoistv monotonnosti i vypuklosti pri lokalnoi approksimatsii”, ZhVM i MF, 33:7 (1993), 996–1003 | MR | Zbl

[7] Subbotin Yu. N., Chernykh N. I., “Poryadok nailuchshei splain-approksimatsii nekotorykh klassov funktsii”, Matem. zametki, 7:1 (1970), 31–42 | MR | Zbl

[8] Kostousov K. V., Shevaldin V. T., “Approximation by Local Exponential Splines”, Topol. Math. Control Theory Differ. Equ. Approx. Theory, Proc. of the Steklov Institute of Mathematics, Suppl. 1, 2004, S147–S157 | MR

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

[10] Shevaldin V. T., “Approksimatsiya lokalnymi parabolicheskimi splainami s proizvolnym raspolozheniem uzlov”, Sib. zhurn. vychisl. matematiki / RAN. Sib. otd-nie. — Novosibirsk, 8:1 (2005), 77–88 | MR | Zbl

[11] Sharma A., Tsimbalario I., “Nekotorye lineinye differentsialnye operatory i obobschennye raznosti”, Matem. zametki, 21:2 (1977), 161–172 | MR | Zbl

[12] Popoviciu T., “Sur le reste dans certains formules lineaires d'approximation de l'analyse”, Mathematica (Cluj), 1 (1959), 95–142 | MR | Zbl