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

#### 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.

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