APPROXIMATION BY LOCAL PARABOLIC SPLINES CONSTRUCTED ON THE BASIS OF INTERPOLATION IN THE MEAN

Elena V. Strelkova     (N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences; Ural Federal University, Ekaterinburg, Russian Federation)

Abstract


The paper deals with approximative and form-retaining properties of the local parabolic splines of the form \(S(x)=\sum\limits_j y_j B_2 (x-jh),\) \( (h>0),\) where \(B_2\) is a normalized parabolic spline with the uniform nodes and functionals \(y_j=y_j(f)\) are given for an arbitrary function \(f\) defined on \(\mathbb{R}\) by means of the equalities $$y_j=\frac{1}{h_1}\int\limits_{\frac{-h_1}{2}}^{\frac{h_1}{2}}
f(jh+t)dt \quad (j\in\mathbb{Z}). $$ On the class \(W^2_\infty\) of functions under \(0<h_1\leq 2h\), the approximation error value is calculated exactly for the case of approximation by such splines in the uniform metrics.


Keywords


Local parabolic splines, Approximation, Mean

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References


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DOI: http://dx.doi.org/10.15826/umj.2017.1.007

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