Approximation of the Hilbert transform on $(0,+\infty)$ by using discrete de la Vall\'ee Poussin filtered polynomials

Abstract

In the present paper, is proposed a method to approximate the Hilbert transform of a given function $f$ on $(0,\infty)$ employing truncated de la Vallée discrete polynomials recently studied in [25]. The method generalizes and improves in some sense a method based on truncated Lagrange interpolating polynomials introduced in [24], since is faster convergent and simpler to apply. Moreover, the additional parameter defining de la Vallée polynomials helps to attain better pointwise approximations. Stability and convergence are studied in weighted uniform spaces and some numerical tests are provided to asses the performance of the procedure.

Keywords

• Hilbert transform
• discrete de la Vallée-Poussin approximation
• Approximation by polynomials

References

1. B. Bialecki, P. Keast: A sinc quadrature subroutine for Cauchy principal value integrals, J. Comput. Appl. Math., 112 (1-2) (1999), 3–20.
2. G. Criscuolo, G. Mastroianni: Convergenza di formule Gaussiane per il calcolo delle derivate di integrali a valor principale secondo Cauchy, Calcolo, 24 (2) (1987), 179–192.
3. S. B. Damelin, K. Diethelm: Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line, Numer. Funct. Anal. Optim., 22 (1-2) (2001), 13–54.
4. M. C. De Bonis, B. Della Vecchia and G. Mastroianni: Approximation of the Hilbert Transform on the real semiaxis using Laguerre zeros, Jour. of Comput. and Appl. Math., 140 (1-2) (2002), 209–229.
5. M. C. De Bonis, B. Della Vecchia and G. Mastroianni: Approximation of the Hilbert Transform on the real semiaxis using Laguerre zeros, J. Comput. Appl. Math., 140 (2002), 209–229.
6. M. C. De Bonis, G. Mastroianni and M. Viggiano: K-functionals, moduli of smoothness and weighted best approximation on the semiaxis, Functions, Series, Operators, Proceedings of the Alexits Memorial Conference, Budapest, (1999).
7. M. C. De Bonis, D. Occorsio: On the simultaneous approximation of a Hilbert transform and its derivatives on the real semiaxis, Appl. Numer. Math., 114 (2017), 132–153.
8. M. C. De Bonis, D. Occorsio: Error bounds for a Gauss-type quadrature rule to evaluate hypersingular integrals, Filomat, 32 (7) (2018), 2525–2543.

9. K. Diethelm: Uniform convergence of optimal order quadrature rules for Cauchy principal value integrals, J. Comput. Appl. Math., 56 (3) (1994), 321–329.
10. F. Filbir, D. Occorsio and W. Themistoclakis: Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes, Mathematics, 8 (4) (2020), Article ID: 542.
11. I. S. Gradshteyn, I.M. Ryzhik: Table of Integrals, Series, and Products, Alan Jeffrey, Fifth ed., Academic Press, Inc. Boston, MA (1994).
12. T. Hasegawa, T. Torii: An automatic quadrature for Cauchy principal value integrals, Math. Comp., 56 (194) (1991), 741–754.
13. F. King: Hilbert Transforms I & II, Cambridge University Press: Cambridge UK (2009).
14. M. L. Krasnov, A. I. Kiselev and G. L. Makarenko: Integral Equations, Eds. MIR, Moscow (1983).
15. I. K. Lifanov, L.N. Poltavskii and G. M. Vainikko: Hypersingular Integral Equations and their Applications, Chapman & Hall CRC 2003.
16. G. Mastroianni: Polynomial inequalities, functional spaces and best approximation on the real semiaxis with Laguerre weights, Electron. Trans. Numer. Anal., 14 (2002), 142–151.
17. G. Mastroianni, G.V. Milovanoviˇc: Some numerical methods for second kind Fredholm integral equation on the real semiaxis, IMA J. Numer. Anal., 29 (4) (2009), 1046–1066.
18. G. Mastroianni, G. Monegato: Truncated quadrature rules over (0,∞) and Nyström type methods, SIAM Jour. Num. Anal., 41 (5) (2003), 1870–1892.
19. G. Mastroianni, D. Occorsio: Lagrange interpolation at Laguerre zeros in some weighted uniform spaces, Acta Math. Hungar., 91 (1-2) (2001), 27–52.
20. S. G. Mikhlin, S. Prössdorf: Singular Integral Operator, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1986).
21. G. Monegato: The Numerical Evaluation of One-Dimensional Cauchy Principal Value Integrals, Computing, 29 (1982), 337–354,
22. I. Notarangelo: Approximation of the Hilbert transform on the real line using Freud weights, Proceedings of the International Conference “Approximation & Computation", dedicated to the 60−th anniversary of G. Milovanoviˇc, Niš August 25–29, 2008.
23. D. Occorsio, M. G. Russo andW. Themistoclakis, Some numerical applications of generalized Bernstein operators, Constr. Math. Anal., 4 (2) (2021), 186–214.
24. D. Occorsio: A method to evaluate the Hilbert transform on (0,+∞), Applied Mathematics and Computation, 217 (12) (2011), 5667–5679.
25. D. Occorsio, W. Themistoclakis: De la Vallée Poussin filtered polynomial approximation on the half line, Appl. Math. Comput., 207 (2025), 569–584.
26. D. Occorsio, W. Themistoclakis: Approximation of the Hilbert transform on the half-line, Appl. Numer. Math., 205 (2024), 101–119.
27. G. Szegö: Orthogonal Polynomials, American Mathematical Society, Providence, RI, 4th Ed. (1975).
28. W. Themistoclakis: Uniform approximation on [−1, 1] via discrete de la Vallée Poussin means, Numer. Algor., 60 (2012), 593–612.