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Approximation of the Hilbert Transform on the real line using Hermite zeros

Authors: M. C. De Bonis, B. Della Vecchia and G. Mastroianni
Journal: Math. Comp. 71 (2002), 1169-1188
MSC (2000): Primary 65D30, 41A05
Published electronically: October 25, 2001
MathSciNet review: 1898749
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Abstract: The authors study the Hilbert Transform on the real line. They introduce some polynomial approximations and some algorithms for its numerical evaluation. Error estimates in uniform norm are given.

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  • 1. Akhiezer N.I.: Lectures on Integral Transforms, AMS, 70 (Translations of Math. Mon.) (1988). MR 89i:44001
  • 2. Bari N.: Treatise on Trigonometric Series, Pergamon Press, New York (1964). MR 30:1347
  • 3. Bialecki B.: Sinc Quadratures for Cauchy Principal Value Integrals, Numerical Integration, Recent Developments, Software and Applications, edited by T.O. Espelid and A. Genz, NATO ASI Series, Series C: Mathematical and Physical Sciences, 357, Kluwer Academic Publishers, Dordrecht (1992). MR 93m:65030
  • 4. Capobianco M.R., Mastroianni G., Russo M.G.: Pointwise and Uniform Approximation of the Finite Hilbert Transform, Proceedings of ICAOR, (Romania) Cluj-Napoca, July 29-August 1, 1 (1996). MR 99b:65168
  • 5. Criscuolo G., Mastroianni G.: On the convergence of an interpolatory product rule for evaluating Cauchy principal value integrals, Math. Comp. 48 (1987), 725-735. MR 88m:65038
  • 6. Criscuolo G., Mastroianni G.: On the convergence of product formulas for the numerical evaluation of derivatives of Cauchy principal value integrals, SIAM J. Numerical Analysis 25 (1988), 713-727. MR 90b:65032
  • 7. Criscuolo G., Mastroianni G.: On the Uniform Convergence of Gaussian Quadrature Rules for Cauchy Principal Value Integrals, Numer. Math. 54 (1989), 445-461. MR 90h:65023
  • 8. Criscuolo G., Della Vecchia B., Lubinsky D.S., Mastroianni G.: Function of the Second Kind for Freud Weights and Series Expansions of Hilbert Transforms, Journal of Mathematical Analysis and Applications, 189 (1995), 256-296. MR 96b:42028
  • 9. Davis P.J., Rabinowitz P.: Methods of Numerical Integration, Academic Press, Inc., (1984). MR 86d:65004
  • 10. De Bonis M.C., Russo M.G.: Computation of the Cauchy Principal Value Integrals on the Real Line, Proceedings of the Workshop ``Advanced Special Functions and Applications", eds. D. Cocolicchio, G. Dattoli and H.M. Srivastava (ARACNE, Rome, 1999).
  • 11. Ditzian Z., Totik V.: Moduli of Smoothness, SCM, Springer-Verlag, New York Berlin Heidelberg London Paris Tokyo, 9 (1987). MR 89h:41002
  • 12. Gautschi W.: A survey of Gauss-Christoffel quadrature formulae, in E.B. Christoffel, The Influence of his Work on Mathematics and Physical Sciences, Birkhäuser, Basel 1981. P.L. Butzer and F. Fehér, eds, 72-147. MR 83g:41031
  • 13. Hunter D.B.: Some Gauss-Type Formulae for the evaluation of Cauchy Principal Values of Integrals, Numer. Math. 19 (1972), 419-424. MR 47:7899
  • 14. Khavin V.P., Nikolski N.K.: Commutative harmonic analysis I, Encyclopedia of Mathematical Sciences, 15 Springer-Verlag, Berlin (1991). MR 93b:42001
  • 15. Kumar S.: A note on quadrature formulae for Cauchy principal value integrals, Journal of the Institute of Mathematics and Its Applications 26 (1980), 447-451. MR 82d:65027
  • 16. Kress V.R., Martensen E.: Anwendung der Rechteckregel auf die reelle Hilberttransformation mit unendlichem Intervall, ZAMM 50, T 61-T 64 (1970). MR 43:8240
  • 17. Levin A.L., Lubinsky D.S.: Christoffel Functions, Orthogonal Polynomials and Nevai's Conjecture for Freud Weights, Constr. Approx. 8 (1992), 463-535. MR 94f:42030
  • 18. Longman I.M.: On the numerical evaluation of Cauchy principal value integrals, Math. Comp. 12 (1958), 205-207. MR 20:6789
  • 19. Lubinsky D.S., Sidi A.: Convergence of product integration rules for functions with interior and endpoint singularities over bounded and unbounded intervals, Tech. Rep. No 215. Computer Science Dept., Technion, Haifa, Israel, 1981; Math. Comp. 46 (1986), no. 173, 229-245. MR 87j:41072
  • 20. Mastroianni G.: On the Convergence of Product Formulas for the Evaluation of certain Two-Dimensional Cauchy Principal Value Integrals, Math. Comp. 52 (1989), 95-101. MR 90a:65049
  • 21. Mastroianni G., Monegato G.: Convergence of product integration rules over $(0,\infty) $ for functions with weak singularities at the origin, Mathematics of Computation 64 No 209 (1995), 237-249. MR 95c:65037
  • 22. Mastroianni G., Monegato G.: Nyström interpolants based on zeros of Laguerre polynomials for some Wiener-Hopf equations, IMA Journal of Numerical Analysis (1997) 17, 621-642. MR 98j:45011
  • 23. Mastroianni G., Ricci P.E.: Error Estimates for a class of integral and discrete transforms, Studia Sci. Math. Hungar. 36 (2000), no. 3-4, 291-305. MR 2001h:44004
  • 24. Mastroianni G., Russo M.G.: Lagrange Interpolation in Weighted Besov Spaces, Const. Approx. (1999), 15, 257-289. MR 2000b:41001
  • 25. Monegato G.: The Numerical Evaluation of One-Dimensional Cauchy Principal Value Integrals, Computing 29 (1982), 337-354. MR 84c:65044
  • 26. Monegato G.: Convergence of Product Formulas for the Numerical Evaluation of certain Two-Dimensional Cauchy Principal Value Integrals, Numer. Math. 43 (1984), 161-173. MR 85h:65049
  • 27. Mastronardi N., Occorsio D.: Some Numerical Algorithms to evaluate Hadamard Finite-Part Integrals, J. Comput. Appl. Math. Vol. 70 (1996), 75-93. MR 97h:65019
  • 28. Mikhlin S.G., Prössdorf S.: Singular Integral Operator, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, (1986). MR 88e:47097
  • 29. Nevai P.G.: Mean Convergence of Lagrange Interpolation II, J. Approx. Theory Vol. 30 (1980), 263-276. MR 82i:41003
  • 30. Poiani E.L.: Mean Cesaro Summability of Laguerre and Hermite Series, Transactions AMS, 173 (1972), 1-31. MR 46:9635
  • 31. Poppe G.P.M., Wijers M.J.: Algorithm 680: Evaluation of the Complex Error Function, ACM Trans. Math. Software 16 (1990), pag. 47. MR 91h:65068b
  • 32. Sklyarov V.P.: On the convergence of Lagrange-Hermite interpolation for unbounded functions, Anal. Math, 20 (1994), 295-308 (in Russian). MR 95h:41006
  • 33. Sloan I.H., Smith W.E.: Properties of interpolatory product integration rules, SIAM J.Numer. Anal. 19 (1982), 427-442. MR 83e:41032
  • 34. Smith W.E., Sloan I.H., Opie A.H.: Product Integration Over Infinite Intervals I. Rules Based on the Zeros of Hermite Polynomials, Mathematics of Computations 40 No 162 (1983), 519-535. MR 85a:65047
  • 35. Stenger F.: Approximations via Whittaker's cardinal function, J. Approx. Theory, 17 (1976), 222-240. MR 58:1885
  • 36. Stenger F.: Numerical Methods based on Whittaker cardinal or Sinc Functions, SIAM Review, 23 (1981), 165-224. MR 83g:65027
  • 37. Stenger F.: Numerical Methods based on Sinc and Analytic Functions, Springer-Verlag, (1993). MR 94k:65003
  • 38. Stenger F.: Summary of Sinc Approximation, preprint.
  • 39. Szabados J.: Weighted Lagrange and Hermite-Fejér interpolation on the real line, J. of Inequal. & Appl., 1 (1997), 99-123. MR 2000h:41010

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Additional Information

M. C. De Bonis
Affiliation: Dipartimento di Matematica, Università della Basilicata, C/da Macchia Romana 85100 Potenza, Italy

B. Della Vecchia
Affiliation: Dipartimento di Matematica, Istituto G. Castelnuovo, Università di Roma La Sapienza, P.le Aldo Moro 2, 00185 Roma, Italy

G. Mastroianni
Affiliation: Dipartimento di Matematica, Università della Basilicata, C/da Macchia Romana 85100 Potenza, Italy

Keywords: Hilbert Transform, orthonormal polynomials, Gaussian quadrature rules, product quadrature rules
Received by editor(s): April 9, 1998
Received by editor(s) in revised form: December 8, 1999, May 12, 2000, and August 18, 2000
Published electronically: October 25, 2001
Additional Notes: This work was supported by M.U.R.S.T. (ex. 40%)
Article copyright: © Copyright 2001 American Mathematical Society

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