Abstract
The convergence rate of histopolation on an interval with combined splines of class C1 having linear/linear rational or quadratic polynomial pieces is studied. The function to histopolate may have finite number of derivative zeros and established convergence rate depends mainly on the behaviour of the derivative near its zeros. Given numerical results are completely consistent with theoretical ones.
Highlights
It is known that the histopolation problem could be reduced to the solution of a corresponding interpolation problem, see, e.g., [14]
E.g., the derivative of the quadratic/linear function as a piece of convexity preserving rational spline is not linear/linear function which is a natural piece of monotonicity preserving rational spline
The main purpose of this paper is to find out the convergence rate of histopolating combined splines consisting of linear/linear rational or quadratic polynomial pieces when the function to histopolate is not necessarily monotone
Summary
It is known that the histopolation problem could be reduced to the solution of a corresponding interpolation problem, see, e.g., [14]. The convergence rates of polynomial spline interpolants and their derivatives are well known. Polynomial spline interpolants and histopolants do not preserve geometric properties like positivity, monotonicity, convexity of the function to approximate. In [3] we established the convergence rate of linear/linear rational spline histopolants in case of strict (uniform) monotonicity. The main purpose of this paper is to find out the convergence rate of histopolating combined splines consisting of linear/linear rational or quadratic polynomial pieces when the function to histopolate is not necessarily monotone. In our paper we will follow the main ideas, representation and basic equations from [2, 4] for histopolating linear/linear rational or combined splines. The most complicated problem in our research occurs to be the analysis of basic equations determining spline parameters
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