Lower estimates on the saturation order of approximation of twice continuously differentiable functions by piecewise constants on convex partitions

O.V. Kozynenko

doi:10.15421/241805

Lower estimates on the saturation order of approximation of twice continuously differentiable functions by piecewise constants on convex partitions

O.V. Kozynenko (Oles Honchar Dnipro National University)

Abstract

We consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants. We show that the saturation order of piecewise constant approximation in $$$L_p$$$ norm on convex partitions with $$$N$$$ cells is $$$N^{-2/(d+1)}$$$, where $$$d$$$ is the number of variables.

Keywords

twice continuously differentiable functions; piecewise constant splines; saturation order; convex partitions

MSC 2020

41A40; 41A25; 46E35

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References

Birman M.S., Solomyak M.Z. "Piecewise polynomial approximation of functions of the classes $$$W^{\alpha}_p$$$", Matem. Sbornik, 1967; 73 (115, 3): pp. 331-355 (in Russian). English translation: Math. USSR-Sb., 1967; 2(3): pp. 295-317.

Davydov O. "Algorithms and error bounds for multivariate piecewise constant approximation", Approximation Algorithms for Complex Systems (E.H. Georgoulis, A. Iske, and J. Levesley, eds.), Springer Proceedings in Mathematics, 2011; 3: pp. 27-45. doi:10.1007/978-3-642-16876-5_2

Davydov O. "Approximation by piecewise constants on convex partitions", J. Approx. Theory, 2012; 164: pp. 346-352. doi:10.1016/j.jat.2011.11.001

Kochurov A.S. "Approximation by piecewise constant functions on the square", East J. Approx., 1995; 1: pp. 463-478.

Kozynenko O., Davydov O., Skorokhodov D. "Adaptive anisotropic approximation of multivariate functions by piecewise constants", International Conference of Young Mathematicians dedicated to the 100th Anniversary of Academician of National Academy of Sciences of Ukraine, Professor Yu. O. Mitropolskiy (1917–2008): int. conf. (Kyiv, Ukraine, June 7-10, 2017), Abstracts, Kyiv: Institute of Mathematics of NAS of Ukraine, 2017; p. 34.

Kudryavcev L.D. A brief course of mathematical analysis, Moscow, Fizmatlit, 2005; 424 p.

Korneichuk N.P. Exact constants in approximation theory, Nauka, Moscow, 1987; 424 p. (in Russian)