On Error Estimates of Local Approximation by Splines
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RROR ESTIMATES OF LOCAL APPROXIMATION BY SPLINES Yu. S. Volkov and V. V. Bogdanov
UDC 517.518.85
Abstract: We consider the so-called simplest formula for local approximation by polynomial splines of order n (Schoenberg splines). The spline itself and all derivatives except that of the highest order, approximate a given function and its corresponding derivatives with the second order. We show that the jump of the highest derivative of order n − 1; i. e., the value of discontinuity, divided by the meshsize, approximates the nth derivative of the original function. We found an asymptotic expansion of the jump. DOI: 10.1134/S0037446620050031 Keywords: local splines, Schoenberg approximation, error estimation, asymptotic expansion
1. Splines, in particular cubic splines, are currently the main tool in the problems related to approximation of functions, especially in applications. It is known [1, 2] that in the classical interpolation problem by a cubic spline of class C 2 it is required to solve some system of linear equations. Such a spline approximates the interpolated function to the fourth order on an arbitrary nonuniform grid. If we use the spline representation in the basis of B-splines, then from the system of equations we find the expansion coefficients that are close to the interpolated values. This circumstance leads to the idea of specifying the coefficients of B-spline expansion by explicit formulas rather than from the interpolation conditions, i. e., to proceed without solving the system of equations. These splines are usually called local splines; the coefficients are specified by the values of the approximated function and its derivatives in some small neighborhood of the knot, for instance, in the form of some functionals that are linear combinations of the function values at the nodes of the mesh. The simplest local approximation formula is a formula whose coefficients are just equal to the values of the original function at the knots of the mesh. The resulting spline does not traverse the given values of the function any more but still approximates the original function. Of course, even on a uniform mesh the order of approximation is only 2; and on a nonuniform mesh, it is 1. But in some problems this accuracy suffices. Methods of local approximation became a standard tool of approximation theory and numerical analysis mostly as a useful alternative to interpolation. Their main advantage is that we have no need to solve any system of linear equations, which differ from interpolation. Although the properties of systems of equations for interpolation splines are already studied [3–5], we get the problems of setting boundary conditions [6–8]. Moreover, the matrices of these systems could be ill-conditioned on significantly nonuniform meshes [9, 10]. The local approximation does not necessarily lead to a loss of accuracy as compared to interpolation. There are some schemes of local approximation where the maximum accuracy order is reached in the same way as in interpolation. The schemes are often called quasi-interpol
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