Abstract
The authors obtain new results on Hermite interpolation based on Jacobi and generalized Jacobi zeros in C1 space and prove error estimates in uniform and weighted Lp norms. The paper gives also the state of art on the topic.
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V. M. Badkov,Convergence in the mean and almost everywhere of Fourier series in polynomials orthogonal on an interval, Math. USSR-Sb. 24 (1974), 223–256.
G. Criscuolo, B. Della Vecchia, G. Mastroianni,Hermite interpolation and mean convergence of its derivatives, Calcolo 1, 2, 28 (1991), 111–127.
G. Criscuolo, B. Della Vecchia, G. Mastroianni Approximation by extended Hermite-Fejer and Hermite interpolation, Colloq. Math. Soc. Janos Bolyai 58 (1991), 151–178.
G. Criscuolo, B. Della Vecchia, G. Mastroianni,Hermite-Fejér and Hermite interpolation, Proceedings of NATO-ASI Conference on Approximation Theory, Spline Functions and Applications, Maratea (ed. by S. P. Singh) (1991), 317–331.
G. Criscuolo, B. Della Vecchia, G. Mastroianni,Extended Hermite interpolation with additional nodes and mean convergence of its derivatives, Rend. Mat. Appl., Serie VII, 12 (1992), 709–728.
G. Criscuolo, B. Della Vecchia, G. Mastroianni,Hermite interpolation on two new nodes matrices, Proceedings of International Conference on Constructive Theory of functions (Varna, Bulgaria) (1992), 93–98.
G. Criscuolo, B. Della Vecchia, G. Mastroianni,Extended Hermite interpolation on Jacobi zeros and mean convergence of its derivatives, Facta Universitatis Nix (1992).
G. Criscuolo, B. Della Vecchia, G. Mastroianni,Approximation by Hermite-Fejer and Hermite interpolation, to appear in Jour. Approx. Theory and its Applications, (1993).
G. Criscuolo, G. Mastroianni,Lagrange interpolation on generalized Jacobi zeros with additional nodes, to appear on Acta Math. Hungar. (1993).
G. Criscuolo, G. Mastroianni,Mean and Uniform Convergence of Quadrature Rules for Evaluating the Finite Hilbert Transform, Progress in Approx. Th. (1991), 141–175.
G. Criscuolo, G. Mastroianni, P. Nevai,Mean Convergence of derivatives of extended Lagrange interpolation with additional nodes, Math. Nachr. 163 (1993).
B. Della Vecchia, G. Mastroianni, P. Vértesi,Simultaneous approximation by Hermite interpolation of higher order, to appear in J. Comp. Appl. Math. (1993).
B. Della Vecchia, G. Mastroianni, P. Vértesi,Weighted L p approximation by Hermite interpolation of higher order plus endpoints, to appear in Studia Sci. Hungar. (1993).
V. K. Dzyadik,Introduction to the theory of uniform approximation of functions by polynomials, Nauka (Moskan, 1977) (in Russian).
I. E. Gopengauz,On a theorem of A.F. Timan on the approximation of functions by polynomials on a finite segment, Math. Notes 1 (1967), 110–116 (in Russian).
P. Nevai,Orthogonal Polynomials, Mem. Amer. Math. Soc.,213, Providence, Rhode Island, 1979.
P. Nevai,Bernstein's inequality in L p,for 0<p<∞, J. Approx. Theory 27 (1979), 239–243.
P. Nevai,Mean convergence of Lagrange interpolation III, Trans. Amer. Math. Soc. 282 (1984), 669–698.
P. Nevai, Y. Xu,Mean convergence of Hermite interpolation, to appear in J. Approx. Theory (1993).
J. Szabados,On the order of magnitude of fundamental polynomials of Hermite interpolation, Acta Math. Hungar. 61 (1993).
G. Szego,Orthogonal Polynomials, Amer. Math. Soc. n. 23, (1939).
P. Vértesi,Remarks on mean convergence of Hermite interpolation, submitted (1992).
P. Vértesi, Y. Xu,Weighted L p convergence of Hermite interpolation of higher order, Acta Math. Hungar. 59 (3–4) (1992), 423–438.
Y. Xu,Norm of the Hermite interpolation operator, Approx. Theory IV, Vol. II, (1989), (ed. by C. K. Chui, L. L. Schumaker, J. D. Ward), 683–686.
Y. Xu,On mean convergence of interpolating polynomials, Colloq. Math. Soc. Jamas Bolyai, 58 (1990), 749–761.
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Vecchia, B.D., Mastroianni, G. The hermite interpolation. Calcolo 30, 371–394 (1993). https://doi.org/10.1007/BF02575884
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DOI: https://doi.org/10.1007/BF02575884