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)=∑jyjB2(x−jh), (h>0), where B2 is a normalized parabolic spline with the uniform nodes and functionals yj=yj(f) are given for an arbitrary function f defined on R by means of the equalities yj=1h1∫−h12h12f(jh+t)dt(j∈Z).On the class W2∞ of functions under 0<h1≤2h, the approximation error value is calculated exactly for the case of approximation by such splines in the uniform metrics.