Approximation by local exponential splines with arbitrary nodes

E. V. Shevaldina

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 9 (2006) no. 4, pp. 391-402

Résumé

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.