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AIMS Mathematics

2023, Volume 8, Issue 9: 20914-20932. doi: 10.3934/math.20231065
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Research article

Bivariate multiquadric quasi-interpolation operators of Lidstone type

  • 1.
    School of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, China
  • 2.
    GongQing Institute Of Science And Technology, Jiujiang 332020, China
  • Received: 13 May 2023 Revised: 14 June 2023 Accepted: 18 June 2023 Published: 30 June 2023

  • MSC : 41A05, 65D05, 65D15

  • In this paper, a kind of bivariate multiquadric quasi-interpolant with the derivatives of a approximated function is studied by combining the known multiquadric quasi-interpolant with the generalized Taylor polynomials that act as the bivariate Lidstone interpolation polynomials. For practical purposes, a kind of improved approximation operator without any derivative of the approximated function is given by using bivariate divided differences to approximate the derivatives. It has the property of high-degree polynomial reproducing. In addition, the improved bivariate quasi-interpolation operators only demand information of the location points rather than the derivatives of the function approximated. Some error bounds in terms of the modulus of continuity of high order and Peano representations for the error are given. Several numerical comparisons with other existing methods are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the algorithm is very simple and easy to implement.

    Citation: Ruifeng Wu. Bivariate multiquadric quasi-interpolation operators of Lidstone type[J]. AIMS Mathematics, 2023, 8(9): 20914-20932. doi: 10.3934/math.20231065

    Related Papers:

  • In this paper, a kind of bivariate multiquadric quasi-interpolant with the derivatives of a approximated function is studied by combining the known multiquadric quasi-interpolant with the generalized Taylor polynomials that act as the bivariate Lidstone interpolation polynomials. For practical purposes, a kind of improved approximation operator without any derivative of the approximated function is given by using bivariate divided differences to approximate the derivatives. It has the property of high-degree polynomial reproducing. In addition, the improved bivariate quasi-interpolation operators only demand information of the location points rather than the derivatives of the function approximated. Some error bounds in terms of the modulus of continuity of high order and Peano representations for the error are given. Several numerical comparisons with other existing methods are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the algorithm is very simple and easy to implement.


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    [1] R. L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 76 (1971), 1905–1915. https://doi.org/10.1029/JB076i008p01905 doi: 10.1029/JB076i008p01905
    [2] R. Franke, Scattered data interpolation: tests of some methods, Math. Comput., 38 (1982), 181–200. https://doi.org/10.2307/2007474 doi: 10.2307/2007474
    [3] C. A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), 11–22.
    [4] M. D. Buhmann, Multivariate Interpolation with Radial Basis Functions, London: University of Cambridge, 1988.
    [5] M. J. D. Powell, Univariate multiquadric approximation: Reproduction of linear polynomials, in: Multivariate Approximation and Interpolation, Basel: Springer, 1990.
    [6] R. K. Beatson, M. J. D. Powell, Univariate multiquadric approximation: Quasi-interpolation to scattered data, Constr. Approx., 8 (1992), 275–288. https://doi.org/10.1007/BF01279020 doi: 10.1007/BF01279020
    [7] Z. M. Wu, R. Schaback, Shape preserving properties and convergence of univariate multiquadric quasi-interpolation, Acta. Math. Appl. Sin. Engl. Ser., 10 (1994), 441–446. https://doi.org/10.1007/BF02016334 doi: 10.1007/BF02016334
    [8] R. H. Wang, M. Xu, A kind of Bernoulli-type quasi-interpolation operator with univariate multiquadrics, Comput. Appl. Math., 29 (2010), 47–60. https://doi.org/10.1590/S1807-03022010000100004 doi: 10.1590/S1807-03022010000100004
    [9] R. H. Wang, M. Xu, Q. Fang, A kind of improved univariate multiquadric quasi-interpolation operators, Comput. Math. Appl., 59 (2010), 451–456. https://doi.org/10.1016/j.camwa.2009.06.023 doi: 10.1016/j.camwa.2009.06.023
    [10] S. Waldron, Increasing the polynomial reproduction of a quasi-interpolation operator, J. Approx. Theory, 161 (2009), 114–126. https://doi.org/10.1016/j.jat.2008.08.011 doi: 10.1016/j.jat.2008.08.011
    [11] C. Rabut, Multivariate divided differences with simple knots, SIAM J. Numer. Anal., 38 (2001), 1294–1311. https://doi.org/10.1137/S003614299935104 doi: 10.1137/S003614299935104
    [12] R. Z. Feng, X. Zhou, A kind of multiquadric quasi-interpolation operator satisfying any degree polynomial reproduction property to scattered data, J. Comput. Appl. Math., 235 (2011), 1502–1514. https://doi.org/10.1016/j.cam.2010.08.037 doi: 10.1016/j.cam.2010.08.037
    [13] G. J. Lidstone, Notes on the extension of Aitken's theorem (for polynomial interpolation) to the Everett types, Proc. Edinb. Math. Soc., 2 (1929), 16–19. https://doi.org/10.1017/S0013091500007501 doi: 10.1017/S0013091500007501
    [14] R. F. Wu, H. L. Li, T. R. Wu, Univariate Lidstone-type multiquadric quasi-interpolants, Comput. Appl. Math., 39 (2020), 141. https://doi.org/10.1007/s40314-020-01159-x doi: 10.1007/s40314-020-01159-x
    [15] T. Cǎtinaş, The combined Shepard-Lidstone bivariate operator, In: Trends an Applications in Constructive Approximation. International Series of Numerical Mathematics, 151 (2005), 77–89. https://doi.org/10.1007/3-7643-7356-3
    [16] F. A. Costabile, F. Dell'Accio, F. Di Tommaso, Complementary Lidstone interpolation on scattered data sets, Numer. Algor., 67 (2013), 157–180. https://doi.org/10.1007/s11075-012-9659-6 doi: 10.1007/s11075-012-9659-6
    [17] R. Caira, F. Dell'Accio, F. Di Tommaso, On the bivariate Shepard-Lidstone operators, J. Comput. Appl. Math., 236 (2012), 1691–1707. https://doi.org/10.1016/j.cam.2011.10.001 doi: 10.1016/j.cam.2011.10.001
    [18] Z. J. Sun, Y. Y. Gao, High order multiquadric trigonometric quasi-interpolation method for solving time-dependent partial differential equations, Numer. Algor., 2022. https://doi.org/10.1007/s11075-022-01486-6
    [19] F. A. Costabile, F. Dell'Accio, Expansion over a rectangle of real functions in Bernoulli polynomials and applications, BIT Numer. Math., 41 (2001), 451–464. https://doi.org/10.1023/A:1021958910686 doi: 10.1023/A:1021958910686
    [20] F. A. Costabile, F. Di Tommaso, E. Longo, A mixed Lagrange-Bernoulli tensor product expansion on the rectangle with applications, Math. Comput. Simulat., 147 (2019), 73–89. https://doi.org/10.1109/MELECON53508.2022.9842881 doi: 10.1109/MELECON53508.2022.9842881
    [21] L. Ling, Multivariate quasi-interpolation schemes for dimension-splitting multiquadric, Appl. Math. Comput., 161 (2005), 195–209. https://doi.org/10.1016/j.amc.2003.12.022 doi: 10.1016/j.amc.2003.12.022
    [22] R. Z. Feng, X. Zhou, A multivariate multiquadric quasi-interpolation with quadric reproduction, J. Comput. Math., 30 (2012), 311–323. https://doi.org/10.4208/jcm.1111-m3495 doi: 10.4208/jcm.1111-m3495
    [23] R. F. Wu, T. R. Wu, H. L. Li, A family of multivariate multiquadric quasi-interpolation operators with higher degree polynomial reproduction, J. Comput. Appl. Math., 274 (2015), 88–108. https://doi.org/10.1016/j.cam.2014.07.008 doi: 10.1016/j.cam.2014.07.008
    [24] R. Z. Feng, S. Peng, Quasi-interpolation scheme for arbitrary dimensional scattered data approximation based on natural neighbors and RBF interpolation, J. Comput. Appl. Math., 329 (2018), 95–105. https://doi.org/10.1016/j.cam.2017.02.026 doi: 10.1016/j.cam.2017.02.026
    [25] S. L. Zhang, H. Q. Yang, Y. Yang, A multiquadric quasi-interpolations method for CEV option pricing model, J. Comput. Appl. Math., 347 (2019), 1–11. https://doi.org/10.1016/j.cam.2018.03.046 doi: 10.1016/j.cam.2018.03.046
    [26] S. S. Li, Y. Duan, L. B. Li, On the meshless quasi-interpolation methods for solving 2D sine-Gordon euqations, Comput. Appl. Math., 41 (2022), 348. https://doi.org/10.1007/s40314-022-02054-3 doi: 10.1007/s40314-022-02054-3
    [27] R. P. Agarwal, P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and their Applications, Dordrecht: Kluwer Academic Publishers, 1993.
    [28] D. D. Stancu, The remainder of certain linear approximaation formulas in two variables, J. Soc. Indust. Appl. Math. Numer. Anal. Ser. B, 1 (1964), 137–163. https://doi.org/10.1137/0701013 doi: 10.1137/0701013
    [29] A. Sard, Linear Approximation, New York: AMS, 1963.
    [30] R. A. Devore, G. G. Lorentz, Constructive Approximation, Berlin: Springer, 1993.
    [31] Z. Ditzian, V. Totik, Moduli of Smoothness, Berlin: Springer, 1987.
    [32] R. J. Renka, A. K. Cline, A triangle-based $C^1$ interpolation method, Rocky Mt. J. Math., 14 (1984), 223–237. https://doi.org/10.1216/RMJ-1984-14-1-223 doi: 10.1216/RMJ-1984-14-1-223
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