Geodesic
Approximation by local $L$-splines corresponding to a linear differential operator of the second order
Trudy Instituta matematiki i mehaniki, Control, stability, and inverse problems of dynamics, Tome 12 (2006) no. 2, pp. 195-213 
V. T. Shevaldin. Approximation by local $L$-splines corresponding to a linear differential operator of the second order. Trudy Instituta matematiki i mehaniki, Control, stability, and inverse problems of dynamics, Tome 12 (2006) no. 2, pp. 195-213. http://geodesic.mathdoc.fr/item/TIMM_2006_12_2_a16/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

For the class of functions $W_\infty^{\mathcal L_2}=\{f:f'\in AC,\|\mathcal L_2(D)f\|_\infty\le1\}$, where $\mathcal L_2(D)$ is a linear differential operator of the second order whose characteristic polynomial has only real roots, we construct a noninterpolating linear positive method of exponential spline approximation possessing extremal and smoothing properties and locally inheriting the monotonicity of the initial data (the values of a function $f\in W_\infty^{\mathcal L_2}$ at the points of a uniform grid). The approximation error is calculated exactly for this class of functions in the uniform metric.

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