Uniform Weighted Approximation by Multivariate Filtered Polynomials

  • Donatella OccorsioEmail author
  • Woula Themistoclakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11973)


The paper concerns the weighted uniform approximation of a real function on the \(d-\)cube \([-1,1]^d\), with \(d>1\), by means of some multivariate filtered polynomials. These polynomials have been deduced, via tensor product, from certain de la Vallée Poussin type means on \([-1,1]\), which generalize classical delayed arithmetic means of Fourier partial sums. They are based on arbitrary sequences of filter coefficients, not necessarily connected with a smooth filter function. Moreover, in the continuous case, they involve Jacobi–Fourier coefficients of the function, while in the discrete approximation, they use function’s values at a grid of Jacobi zeros. In both the cases we state simple sufficient conditions on the filter coefficients and the underlying Jacobi weights, in order to get near–best approximation polynomials, having uniformly bounded Lebesgue constants in suitable spaces of locally continuous functions equipped with weighted uniform norm. The results can be useful in the construction of projection methods for solving Fredholm integral equations, whose solutions present singularities on the boundary. Some numerical experiments on the behavior of the Lebesgue constants and some trials on the attenuation of the Gibbs phenomenon are also shown.


Weighted polynomial approximation de la Vallée Poussin means Filtered approximation Lebesgue constants Projection methods for singular integral equations Gibbs phenomenon 



The authors are grateful to the anonymous referees for carefully reviewing this paper and for their valuable comments and suggestions.

This research has been accomplished within Rete ITaliana di Approssimazione (RITA) and partially supported by the GNCS-INdAM funds 2019, project “Discretizzazione di misure, approssimazione di operatori integrali ed applicazioni”.


  1. 1.
    Bos, L., Caliari, M., De Marchi, S., Vianello, M.: A numerical study of the Xu polynomial interpolation formula in two variables. Computing 76, 311–324 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bos, L., Caliari, M., De Marchi, S., Vianello, M., Xu, Y.: Bivariate Lagrange interpolation at the Padua points: the generating curve approach. J. Approx. Theory 143, 15–25 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caliari, M., De Marchi, S., Vianello, M.: Bivariate Lagrange interpolation at the Padua points: computational aspects. J. Comput. Appl. Math. 221, 284–292 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Capobianco, M.R., Criscuolo, G., Junghanns, P., Luther, U.: Uniform convergence of the collocation method for Prandtl’s integro-differential equation. ANZIAM J. 42(1), 151–168 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Capobianco, M.R., Themistoclakis, W.: Interpolating polynomial wavelets on \([-1, 1]\). Adv. Comput. Math. 23(4), 353–374 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    De Bonis, M.C., Occorsio, D.: Quadrature methods for integro-differential equations of Prandtl’s type in weighted uniform norms. AMC (to appear)Google Scholar
  7. 7.
    De Bonis, M.C., Occorsio, D.: On the simultaneous approximation of a Hilbert transform and its derivatives on the real semiaxis. Appl. Numer. Math. 114, 132–153 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    De Marchi, S., Erb, W., Marchetti, F.: Spectral filtering for the reduction of the gibbs phenomenon for polynomial approximation methods on Lissajous curves with applications in MPI. Dolom. Res. Notes Approx. 10, 128–137 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fermo, L., Russo, M.G., Serafini, G.: Numerical methods for Cauchy bisingular integral equations of the first kind on the square. J. Sci. Comput. 79(1), 103–127 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Filbir, F., Mhaskar, H.N., Prestin, J.: On a filter for exponentially localized kernels based on Jacobi polynomials. J. Approx. Theory 160, 256–280 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Filbir, F., Themistoclakis, W.: On the construction of de la Vallée Poussin means for orthogonal polynomials using convolution structures. J. Comput. Anal. Appl. 6, 297–312 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gottlieb, D., Shu, C.-W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644–668 (1997)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mhaskar, H.N.: Localized summability kernels for Jacobi expansions. In: Rassias, T.M., Gupta, V. (eds.) Mathematical Analysis, Approximation Theory and Their Applications. SOIA, vol. 111, pp. 417–434. Springer, Cham (2016). Scholar
  14. 14.
    Mastroianni, G., Milovanovic, G.: Interpolation Processes. Basic Theory and Applications. Springer Monographs in Mathematics. Springer, Heidelberg (2008). Scholar
  15. 15.
    Mastroianni G., Milovanovi\(\acute{c}\), G., Occorsio, D.: A Nyström method for two variables Fredholm integral equations on triangles. Appl. Math. Comput. 219, 7653–7662 (2013)Google Scholar
  16. 16.
    Mastroianni, G., Russo, M.G., Themistoclakis, W.: The boundedness of the Cauchy singular integral operator in weighted Besov type spaces with uniform norms. Integr. Eqn. Oper. Theory 42, 57–89 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mastroianni, G., Themistoclakis, W.: A numerical method for the generalized airfoil equation based on the de la Vallée Poussin interpolation. J. Comput. Appl. Math. 180, 71–105 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Occorsio, D., Russo, M.G.: Numerical methods for Fredholm integral equations on the square. Appl. Math. Comput. 218(5), 2318–2333 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Occorsio, D., Russo, M.G.: Nyström methods for Fredholm integral equations using equispaced points. Filomat 28(1), 49–63 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Occorsio, D., Themistoclakis, W.: Uniform weighted approximation on the square by polynomial interpolation at Chebyshev nodes. Submitted to a Special Iusse of NUMTA 2019Google Scholar
  21. 21.
    Sloan, I.H., Womersley, R.S.: Filtered hyperinterpolation: a constructive polynomial approximation on the sphere. Int. J. Geomath. 3, 95–117 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Themistoclakis, W.: Weighted \(L^1\) approximation on \([-1,1]\) via discrete de la Vallée Poussin means. Math. Comput. Simul. 147, 279–292 (2018)CrossRefGoogle Scholar
  23. 23.
    Themistoclakis, W.: Uniform approximation on \([-1,1]\) via discrete de la Vallée Poussin means. Numer. Algorithms 60, 593–612 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Themistoclakis, W., Van Barel, M.: Generalized de la Vallée Poussin approximations on \([-1,1]\). Numer. Algorithms 75, 1–31 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Themistoclakis, W., Vecchio, A.: On the solution of a class of nonlinear systems governed by an M -matrix. Discrete Dyn. Nat. Soc. 2012, 12 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Xu, Y.: Lagrange interpolation on Chebyshev: points of two variables. J. Approx. Theory 87(2), 220–238 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematics, Computer Science and EconomicsUniversity of BasilicataPotenzaItaly
  2. 2.C.N.R. National Research Council of Italy, I.A.C. Istituto per le Applicazioni del Calcolo “Mauro Picone”NapoliItaly

Personalised recommendations