Abstract
In order to approximate continuous functions on \([0,+\infty )\), we consider a Lagrange–Hermite polynomial, interpolating a finite section of the function at the zeros of some orthogonal polynomials and, with its first \((r-1)\) derivatives, at the point 0. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator. Moreover, we prove optimal estimates for the error of this process in the weighted \(L^p\) and uniform metric.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balázs, K.: Lagrange and Hermite interpolation processes on the positive real line. J. Approx. Theory 50(1), 18–24 (1987)
Cvetkovic, A.S., Milovanovic, G.V.: The mathematica package “Orthogonal Polynomials”. Facta Univ. Ser. Math. Inform. 19, 17–36 (2004)
Ditzian, Z., Totik, V.: Moduli of Smoothness, Springer Series in Computational Mathematics 9. Springer, New York (1987)
Erdős, P., Turán, P.: On interpolation. III. Interpolatory theory of polynomials. Ann. Math. (2) 41, 510–553 (1940)
Freud, G.: Convergence of the Lagrange interpolation on the infinite interval. Mat. Lapok 18, 289–292 (1967)
Frammartino, C.: A Nyström method for solving a boundary value. In: Gautchi, W., Mastroianni, G., Rassias, T. (eds.) Approximation and Computation. Springer Optim. Appl. 42, pp. 311–325. Springer, New York (2011)
Kasuga, T., Sakai, R.: Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121, 13–53 (2003)
Laurita, C., Mastroianni, G.: \(L^p\)-Convergence of lagrange interpolation on the semiaxis. Acta Math. Hungar. 120(3), 249–273 (2008)
Levin, A.L., Lubinsky, D.S.: Christoffel functions, orthogonal polynomials, and Nevai’s conjecture for Freud weights. Constr. Approx. 8(4), 463–535 (1992)
Levin, A.L., Lubinsky, D.S.: Orthogonal Polynomials for Exponential Weights, CSM Books in Mathematics/Ouvrages de Mathématiques de la SMC, 4. Springer, New York (2001)
Levin, A.L., Lubinsky, D.S.: Orthogonal polynomials for weights \(x^{2\rho } e^{-2Q(x)}\) on \([0, d)\). J. Approx. Theory 134(2), 199–256 (2005)
Levin, A.L., Lubinsky, D.S.: Orthogonal polynomials for weights \(x^{2\rho } e^{-2Q(x)}\) on \([0, d)\), II. J. Approx. Theory 139(1), 107–143 (2006)
Lubinsky, D.S., Mastroianni, G.: Mean convergence of extended Lagrange interpolation with Freud weights. Acta Math. Hungar. 84(1–2), 47–63 (1999)
Lubinsky, D.S., Mastroianni, G.: Converse quadrature sum inequalities for Freud weights. II. Acta Math. Hungar. 96(1–2), 147–168 (2002)
Mastroianni, G., Milovanović, G.V.: Interpolation Processes. Basic Theory and Applications, Springer Monographs in Mathematics. Springer, Berlin (2008)
Mastroianni, G., Milovanovic, G.V.: Some numerical methods for second kind Fredholm integral equation on the real semiaxis. IMA J. Numer. Anal. 29, 1046–1066 (2009)
Mastroianni, G., Milovanović, G.V., Notarangelo, I.: On an interpolation process of Lagrange-Hermite type. Publication de l’Institut Mathématique Nouvelle série tome 91(105), 163–175 (2012)
Mastroianni, G., Notarangelo, I.: A Lagrange-type projector on the real line. Math. Comput. 79(269), 327–352 (2010)
Mastroianni, G., Notarangelo, I.: Some Fourier-type operators for functions on unbounded intervals. Acta Math. Hungar. 127(4), 347–375 (2010)
Mastroianni, G., Occorsio, D.: Lagrange interpolation at Laguerre zeros in some weighted uniform spaces. Acta Math. Hungar. 91(1–2), 27–52 (2001)
Mastroianni, G., Szabados, J.: Polynomial approximation on the real semiaxis with generalized Laguerre weights. Stud. Univ. Babes-Bolyai LII(Number 4), 89–103 (2007)
Mastroianni, G., Vértesi, P.: Fourier sums and Lagrange interpolation on \((0,+\infty )\) and \((-\infty,+\infty )\). In: Govil, N.K., Mhaskar, H.N., Mohapatra, R.N., Nashed, Z., Szabados, J. (eds.) Frontiers in Interpolation and Approximation, Dedicated to the Memory of A. Sharma, pp. 307–344. Taylor & Francis Books, Boca Raton, Florida (2006)
Muckenhoupt, B., Wheeden, L.: Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform. Stud. Math. 55(3), 279–294 (1976)
Nevai, P.: On Lagrange interpolation based on Laguerre roots [Hungarian]. Mat. Lapok 22, 149–164 (1971)
Nevai, P.: On the convergence of Lagrange interpolation based on Laguerre roots, [Russian]. Publ. Math. 20, 235–239 (1973)
Nevai, P.: Lagrange interpolation at zeros of orthogonal polynomials. In: Lorentz, G.G. (ed.) Approximation Theory, pp. 163–201. Academic Press, New York (1976)
Shen, J., Wang, L.-L.: Some recent advances on spectral methods for unbounded domains. Commun. Comput. Phys. 5(2–4), 195–241 (2009)
Szabados, J.: Weighted Lagrange and Hermite-Fejér interpolation on the real line. J. Inequal. Appl. 1(2), 99–123 (1997)
Acknowledgments
The authors would like to thank the referees for carefully reading the manuscript and for their suggestions which contributed in improving the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author and the third authors were partially supported by University of Basilicata (local funds). The second author was supported by University of Basilicata (local funds) and by National Group of Computing Science GNCS–INDAM.
Rights and permissions
About this article
Cite this article
Mastroianni, G., Notarangelo, I. & Pastore, P. Lagrange–Hermite interpolation on the real semiaxis. Calcolo 53, 235–261 (2016). https://doi.org/10.1007/s10092-015-0147-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10092-015-0147-y
Keywords
- Hermite–Lagrange interpolation
- Approximation by algebraic polynomials
- Orthogonal polynomials
- Generalized Laguerre weights
- Real semiaxis