Abstract
The approximation of functions and their derivatives by local splines of arbitrary degree and deficiency 1 on the basis of the values of a function f on a uniform mesh with step α is considered. Asymptotic expressions are obtained in powers of α for the simplest Schoenberg splines and their derivatives, together with exact estimates of the approximation error. Splines of the minimum pattern of any degree l are obtained, which approximate f ́ (s) ( s arbitrary) with accuracy O( α 1 + 1). Asymptotic expressions are obtained for them. Local splines of arbitrary degree, quasi-interpolating f ́ (s) are also constructed, and asymptotic relations are obtained for them, along with asymptotic relations for the interpolation splines of deficiency 1 of any degree. The results are based on properties of B splines proved in the paper.
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