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Numerical Analysis

Bivariate general Appell interpolation problem

Profile image of Anna NapoliAnna Napoli

2022, Numerical Algorithms

https://doi.org/10.1007/S11075-022-01272-4
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26 pages

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Abstract

In this paper, the solution to a bivariate Appell interpolation problem proposed in a previous work is given. Bounds of the truncation error are considered. Ten new interpolants for real, regular, bivariate functions are constructed. Numerical examples and comparisons with bivariate Bernstein polynomials are considered.

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References (30)

  1. Anshelevich, M.l.: Appell polynomials and their relatives. Int. Math. Res. Not. 65, 3469-3531 (2004)
  2. Aygunes, A.A.: Higher-order Euler-type polynomials and their applications. Fixed Point Theory Appl. 2013(1), 1-11 (2013)
  3. Boas, R.P., Buck, R.C.: Polynomial Expansions of Analytic Functions, vol. 19. Springer Science & Business Media (2013)
  4. Bretti, G., Cesarano, C.: Ricci, P.E. Laguerre-type exponentials and generalized Appell polynomials. Comput. Math. Appl. 48, 833-839 (2004)
  5. Bretti, G., Natalini, P., Ricci, P.E.: Generalizations of the Bernoulli and Appell polynomials. Abstr. Appl. Anal. 7, 613-623 (2004)
  6. Buhmann, M.D.: Radial basis functions. Acta Numerica 9, 1-38 (2000)
  7. Cesarano, C.: A note on generalized Hermite polynomials. Int. J. Appl. Math. Informat. 8, 1-6 (2014)
  8. Choi, J.: Explicit formulas for Bernoulli polynomials of order n. Indian. J. Pure Appl. Math. 27(7), 667-674 (1996)
  9. Corcino, C.B., Corcino, R.B., Canete, J.A.A.: Some formulae of Genocchi polynomials of higher order. arXiv:2009.03481 (2020)
  10. Costabile, F.A.: Modern Umbral Calculus. An Elementary Introduction with Applications to Linear Interpolation and Operator Approximation Theory, vol. 72. Walter de Gruyter GmbH & Co KG (2019)
  11. Costabile, F.A., Gualtieri, M.I., Napoli, A.: General bivariate Appell polynomials via matrix calculus and related interpolation hints. Mathematics 9(9), 964 (2021)
  12. Costabile, F.A., Gualtieri, M.I., Napoli, A.: Lidstone-Euler interpolation and related high even order boundary value problem. Calcolo 58(25), 1-24 (2021)
  13. Dattoli, G., Cesarano, C., Lorenzutta, S.: Bernoulli numbers and polynomials from a more general point of view. Rend. Math. Appl. 22(7), 193-202 (2002)
  14. Dattoli, G., Germano, B., Ricci, P.E.: Comments on monomiality, ordinary polynomials and associated bi-orthogonal functions. Appl. Math. Comput. 154(1), 219-227 (2004)
  15. Davis, P.J.: Interpolation and Approximation. Courier Corporation (1975)
  16. Dragomir, S.S., Qi, F., Hanna, G.T., Cerone, P.: New Taylor-like expansions for functions of two variables and estimates of their remainders. J. Korean Soc. Ind. Appl. Math. 9(2), 1-15 (2005)
  17. Engels, H.: Numerical Quadrature and Cubature. Academic Press (1980)
  18. Gasca, M., Sauer, T.: Polynomial interpolation in several variables. Adv. Comput. Math. 12(4), 377- 410 (2000)
  19. Gasca, M., Sauer, T.: On the history of multivariate polynomial interpolation. Numerical Analysis: Historical Developments in the 20th Century, 135-147 (2001)
  20. Gould, H.W., Hopper, A.T., et al.: Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29(1), 51-63 (1962)
  21. Khan, S., Raza, N., et al.: General-Appell polynomials within the context of monomiality principle. Int. J. Anal., 328032 (2013)
  22. Kim, D.S., Kim, T.: A note on higher-order Bernoulli polynomials. J. Inequal. Appl. 2013(1), 1-9 (2013)
  23. Lorentz, R.A.: Multivariate Birkhoff Interpolation. Springer (2006)
  24. Lorentz, R.A.: Multivariate Hermite interpolation by algebraic polynomials: A survey. J. Comput. Appl. Math. 122(1-2), 167-201 (2000)
  25. Martinez, F.L.: Some properties of two-dimensional Bernstein polynomials. J. Approx. Theory 59(3), 300-306 (1989)
  26. Nörlund, N.E.: Vorlesungen über Differenzenrechnung, vol. 13 Springer (1924)
  27. Olver, P.J.: On Multivariate Interpolation. Available online. http://www.math.umn.edu/ ∼ olver (1916)
  28. Powell, M.J.D. et al.: Approximation Theory and Methods. Cambridge University Press (1981)
  29. Sard, A.: Linear approximation. American Mathematical Soc (1963)
  30. Xu, Y.: Polynomial interpolation in several variables, cubature formulae, and ideals. Adv. Comput. Math. 12(4), 363-376 (2000)

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