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An iterated quasi-interpolation approach for derivative approximation

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Abstract

Given discrete function values sampled at uniform centers, the iterated quasi-interpolation approach for approximating the m th derivative consists of two steps. The first step adopts m successive applications of the operator DQ (the quasi-interpolation operator Q first, and then the differentiation operator D) to get approximated values of the m th derivative at uniform centers. Then, by one further application of the quasi-interpolation operator Q to corresponding approximated derivative values gives the final approximation of the m th derivative. The most salient feature of the approach is that it approximates all derivatives with the same convergence rate. In addition, it is valid for a general multivariate function, compared with the existing iterated interpolation approaches that are only valid for periodic functions, so far. Numerical examples of approximating high-order derivatives using both the iterated and direct approach based on B-spline quasi-interpolation and multiquadric quasi-interpolation are presented at the end of the paper, which demonstrate that the iterated quasi-interpolation approach provides higher approximation orders than the corresponding direct approach.

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Acknowledgments

We are grateful to the editor and the referee for insightful comments and valuable suggestions that helped us to improve the presentation of the manuscript.

Funding

This work is supported by NSFC (11871074, 11501006, 61672032), NSFC Key Project (11631015,91330201), Hong Kong Scholars Program (2018046), SGST (12DZ 2272800), Fund of Introducing Leaders of Science and Technology of Anhui University (J10117700057), the 4th Project of Cultivating Backbone of Young Teachers of Anhui University (J01005138), and Anhui Provincial Science and Technology Major Project (16030701091).

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Authors and Affiliations

  1. Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China

    Zhengjie Sun, Zongmin Wu & Wenwu Gao

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Zhengjie Sun

  3. Shanghai Center for Mathematical Sciences, Shanghai, People’s Republic of China

    Zongmin Wu

  4. School of Economics, Anhui University, Hefei, People’s Republic of China

    Wenwu Gao

  5. Anhui Engineering Laboratory of Agro-Ecological Big Data, Anhui University, Hefei, People’s Republic of China

    Wenwu Gao

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  1. Zhengjie Sun
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Correspondence to Wenwu Gao.

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Sun, Z., Wu, Z. & Gao, W. An iterated quasi-interpolation approach for derivative approximation. Numer Algor 85, 255–276 (2020). https://doi.org/10.1007/s11075-019-00812-9

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