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<p>We&#xa0;consider the problem of approximating a&#xa0;function by integral quadratic splines, also known as integro quadratic splines, from the known integral averages over the mesh intervals.It&#xa0;is shown that, in the mean interpolation problem, an&#xa0;integral quadratic spline can be defined in terms of an&#xa0;interpolating cubic spline.Since interpolating cubic splines have been studied quite well, some of their properties can be transferred to integral quadratic splines.We&#xa0;propose formulas for quasi-interpolation by integral quadratic splines.The&#xa0;problem of local approximation by integral splines is considered for the first time on an&#xa0;arbitrary nonuniform mesh.On&#xa0;a uniform mesh, we find all superconvergence points for mean interpolation and quasi-interpolation, i.e., the points at which the approximation order increases.It&#xa0;is shown that the jump (the ratio of the magnitude of the discontinuity to the mesh size) of the second derivative of both splines approximates the third derivative with fourth-order accuracy.</p>

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Formulas for Quasi-Interpolation by Integral Quadratic Splines

  • 期刊论文

  • 发表日期：

    2026年7月30日

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  • Yu. S. Volkov,
  • T. Zhanlav,
  • R.-O. Mijiddorj

摘要

We consider the problem of approximating a function by integral quadratic splines, also known as integro quadratic splines, from the known integral averages over the mesh intervals.It is shown that, in the mean interpolation problem, an integral quadratic spline can be defined in terms of an interpolating cubic spline.Since interpolating cubic splines have been studied quite well, some of their properties can be transferred to integral quadratic splines.We propose formulas for quasi-interpolation by integral quadratic splines.The problem of local approximation by integral splines is considered for the first time on an arbitrary nonuniform mesh.On a uniform mesh, we find all superconvergence points for mean interpolation and quasi-interpolation, i.e., the points at which the approximation order increases.It is shown that the jump (the ratio of the magnitude of the discontinuity to the mesh size) of the second derivative of both splines approximates the third derivative with fourth-order accuracy.