Summary
The authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm. They prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Badkov, V.M. (1974): Convergence in the mean and almost everywhere of Fourier series in polynomials orthogonal on an interval. Math. USSR-Sb.24, 223–256
Balázs, K., Kilgore, T. (1990): Simultaneous approximation of derivatives by Lagrange and Hermite interpolation. J. Approx. Theory60, 231–244
Balázs, K., Kilgore, T. (1989): Simultaneous approximation of derivatives by interpolation. In: C.K. Chui, L.L. Schumaker, J.D. Ward, eds., Approximation Theory VI. Academic Press, New York, pp. 25–27
Balázs, K., Kilgore, T.: Interpolation and simultaneous mean convergence of derivatives, Submitted
Criscuolo, G., Mastroianni, G. (1991): Mean and uniform convergence of quadrature rules for evaluating the finite Hilbert transform. In: P. Nevai, A. Pinkus, eds., Progress in Approximation Theory. Academic Press, New York
Criscuolo, G., Della Vecchia, B., Mastroianni, G.: Approximation by Hermite-Fejér and Hermite interpolation. Submitted
Criscuolo, G., Della Vecchia, B., Mastroianni, G.: Approximation by extended Hermite-Fejér and Hermite interpolation. Submitted
Criscuolo, G., Della Vecchia, B., Mastroianni, G.: Pointwise estimates for extended Hermite interpolating polynomial with additional nodes. Submitted
Criscuolo, G., Della Vecchia, B., Mastroianni, G.: Extended Hermite interpolation on Jacobi zeros and mean convergence of its derivatives. Submitted
Criscuolo, G., Mastroianni, G., Nevai, P.: Mean convergence of derivatives of extended Lagrange interpolation with additional nodes. Submitted
Criscuolo, G., Mastroianni, G., Vértesi, P.: Pointwise simultaneous convergence of the extended Lagrange interpolation with additional knots. Submitted
Dzjadik, V.K. (1977): Introduction to the theory of uniform approximation of functions by polynomials. Nauka, Moscow [in Russian]
Guatschi, W. (1988): Gauss-Kronrod quadrature—A survey. In: G.V. Milovanović, ed., Numerical Methods and Approximation Theory III. Prosveta, Nis, pp. 39–66
Gopengauz, I.E. (1967): On a theorem of A.F. Timan on the approximation of functions by polynomials on a finite segment. Math. Notes1, 110–116 [in Russian]
Kilgore, T., Prestin, J.: On the order of convergence of simultaneous approximation by Lagrange-Hermite interpolation. Submitted
Mastroianni, G.: Uniform convergence of Lagrange interpolation. Submitted
Mastroianni, G., Nevai, P. (1991): Mean convergence of Lagrange interpolation. J. Comput. Appl. Math. (to appear)
Mastroianni, G., Prössdorf, S.: A quadrature method for Cauchy integral equations with weakly singular perturbation kernel. Submitted
Mastroianni, G., Vértesi, P.: Simultaneous pointwise approximation of Lagrange interpolation. Submitted
Maté, A., Nevai, P., Totik, V. (1987): Extensions of Szwgö's theory of orthogonal polynomials. Lagrange interpolation. Constr. Approx.3, 51–72
Monegato, G. (1982): Stieltjes polynomials and related quadrature rules. SIAM Rev.24, 137–158
Nevai, P. (1979): Orthogonal polynomials. Memoirs Amer. Math. Soc., Vol. 213. Amer. Math. Soc., Providence, Rhode Island
Nevai, P. (1984): Mean convergence of Lagrange interpolation. III. Trans. Amer. Math. Soc.282, 669–698
Nevai, P., Vértesi, P. (1985): Mean convergence of Hermite-Fejér interpolation. J. Math. Anal. Appl.105, 26–58
Nevai, P., Vértesi, P. (1989): Results on mean convergence of derivatives of Lagrange interpolation. In: C.K. Chui, L.L. Schumaker, J.D. Ward, eds., Approximation Theory VI. Academic Press, New York, pp. 491–494
Prasad, J., Varma, A.K. (1981): A study of some interpolating processes based on the roots of Legendre polynomials. J. Approx. Theory31, 244–252
Runck, P.O., Vértesi, P.: Some good systems for derivatives of Lagrange interpolatory operators. Submitted
Srivastava, K.B. (1986): Some results in the theory of interpolation using the Legendre polynomial and its derivative. J. Approx. Theory47, 1–16
Szabados, J. (1991): On the convergence of the derivatives of projection operators. Analysis (to appear)
Turan, P. (1966): On some problems in the theory of the mechanical quadrature. Mathematica (Cluj)8, 181–192
Author information
Authors and Affiliations
Additional information
This material is based upon work supported by the Italian Research Council (first and second authors), and by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica (second and third author).
Rights and permissions
About this article
Cite this article
Criscuolo, G., Mastroianni, G. & Occorsio, D. Uniform convergence of derivatives of extended Lagrange interpolation. Numer. Math. 60, 195–218 (1991). https://doi.org/10.1007/BF01385721
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01385721