Abstract
We study the behavior of some “truncated” Gaussian rules based on the zeros of Pollaczek-type polynomials. These formulas are stable and converge with the order of the best polynomial approximation in suitable function spaces. Moreover, we apply these results to the related Lagrange interpolation process and to prove the stability and the convergence of a Nyström method for Fredholm integral equations of the second kind. Finally, some numerical examples are shown.
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References
Atkinson K.E.: The Numerical Solution of Integral Equations of the second kind, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (1997)
Cvetković A.S., Milovanović G.V.: The Mathematica package “Orthogonal Polynomials”. Facta Univ. Ser. Math. Inform. 19, 17–36 (2004)
Della Vecchia B., Mastroianni G.: Gaussian rules on unbounded intervals, (Oberwolfach, 2001). J. Complex. 19(3), 247–258 (2003)
Erdős T., Turán P.: On interpolation. I. Quadrature- and mean-convergence in the Lagrange-interpolation. Ann. Math. (2) 38, 142–155 (1937)
Freud G.: Orthogonal Polynomials. Akadémiai Kiadó/Pergamon Press, Budapest (1971)
Levin, A.L., Lubinsky, D.S.: Christoffel functions and orthogonal polynomials for exponential weights on [−1, 1]. Mem. Am. Math. Soc. 111(535) (1994)
Levin A.L., Lubinsky D.S.: Orthogonal polynomials for exponential weights, CSM Books in Mathematics, 4. Springer, New York (2001)
Mastroianni G., Milovanović G.V.: Interpolation processes. Basic theory and applications. Springer Monographs in Mathematics. Springer, Berlin (2008)
Mastroianni, G., Monegato, G.: Truncated Gauss–Laguerre quadrature rules. In: Trigiante, D. (ed.) Recent Trends in Numerical Analysism, pp. 1870–1892, Nova Science (2000)
Mastroianni G., Monegato G.: Truncated quadrature rules over ]0, + ∞[ and Nyström type methods. SIAM J. Numer. Anal. 41, 1870–1892 (2003)
Mastroianni G., Notarangelo I.: Polynomial approximation with an exponential weight on [−1, 1] (revisiting some of Lubinsky’s results). Acta Sci. Math. (Szeged) 77(1–2), 73–113 (2011)
Mastroianni G., Notarangelo I.: A Lagrange-type projector on the real line. Math. Comput. 79(269), 327–352 (2010)
Mastroianni, G., Occorsio, D.: Lagrange interpolation based at Sonin–Markov zeros. In: Proceedings of the Fourth International Conference on Functional Analysis and Approximation Theory, vol. II (Acquafredda di Maratea, 2000), Rend. Circ. Mat. Palermo, (2), Suppl. no. 68, pp. 683–697 (2002)
Mastroianni G., Russo M.G.: Lagrange interpolation in weighted Besov spaces. Constr. Approx. 15, 257–289 (1999)
Mastroianni, G., Vértesi, P.: Fourier sums and Lagrange interpolation on (0, + ∞) and (−∞, + ∞), Frontiers in Interpolation and Approximation, Dedicated to the memory of A. Sharma, pp. 307–344, Pure Appl. Math. (Boca Raton), 282, Chapman & Hall/CRC, Boca Raton (2007)
Notarangelo, I.: Polynomial inequalities and embedding theorems with exponential weights on (−1, 1), Acta Math. Hungarica. doi:10.1007/s10474-011-0152-9
Prössdorf S., Silbermann B.: Numerical Analysis for Integral and Related Operator Equations. Akademie Verlag, Birkhäuser Verlag, Berlin, Basel (1991)
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Dedicated to Professor Giovanni Monegato for his 60th birthday.
This research was supported by University of Basilicata (local funds) and by PRIN 2008 “Equazioni integrali con struttura e sistemi lineari” N. 20083KLJEZ.
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De Bonis, M.C., Mastroianni, G. & Notarangelo, I. Gaussian quadrature rules with exponential weights on (−1, 1). Numer. Math. 120, 433–464 (2012). https://doi.org/10.1007/s00211-011-0417-9
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DOI: https://doi.org/10.1007/s00211-011-0417-9