Advertisement

Results in Mathematics

, 74:119 | Cite as

Exponential Sampling Series: Convergence in Mellin–Lebesgue Spaces

  • Carlo BardaroEmail author
  • Ilaria Mantellini
  • Gerhard Schmeisser
Article
  • 83 Downloads

Abstract

In this paper we study norm-convergence to a function f of its generalized exponential sampling series in weighted Lebesgue spaces. Key roles are taken by a result on the norm-density of the test functions and the notion of bounded coarse variation. Some examples are described.

Keywords

Mellin–Lebesgue spaces generalized exponential sampling series bounded coarse variation 

Mathematics Subject Classification

Primary 41A58 42C15 94A20 Secondary 46E22 

Notes

Acknowledgements

Carlo Bardaro and Ilaria Mantellini have been partially supported by the “Gruppo Nazionale per l’Analisi Matematica e Applicazioni (GNAMPA)” of the “Instituto di Alta Matematica (INDAM)” as well as by the projects “Ricerca di Base 2017 of University of Perugia (title: Misura, Integrazione, Approssimazione e loro Applicazioni)” and “Progetto Fondazione Cassa di Risparmio cod. nr. 2018.0419.021 (title: Metodi e Processi di Intelligenza artificiale per lo sviluppo di una banca di immagini mediche per fini diagnostici (B.I.M.))”.

References

  1. 1.
    Angeloni, L., Costarelli, D., Vinti, G.: A characterization of the convergence in variation for the generalized sampling series. Ann. Acad. Sci. Fenn. Math. 43(2), 755–767 (2018)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Angeloni, L., Vinti, G.: A unified approach to approximation results with applications to nonlinear sampling theory. Int. J. Math. Sci. 3(1), 93–128 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Angeloni, L., Vinti, G.: Rate of approximation for nonlinear integral operators with application to signal processing. Differ. Integral Equ. 18(8), 855–890 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bardaro, C., Butzer, P.L., Mantellini, I.: The exponential sampling theorem of signal analysis and the reproduction kernel formula in the Mellin transform setting. Sampl. Theory Signal Image Process. 13(1), 35–66 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bardaro, C., Butzer, P.L., Mantellini, I.: The Mellin–Parseval formula and its interconnections with the exponential sampling theorem of optical physics. Integral Transforms Spec. Funct. 27(1), 17–29 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: On the Paley–Wiener theorem in the Mellin transform setting. J. Approx. Theory 207, 60–75 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Mellin analysis and its basic associated metric-applications to sampling theory. Anal. Math. 42(4), 297–321 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: A fresh approach to the Paley–Wiener theorem for Mellin transforms and the Mellin–Hardy spaces. Math. Nachr. 290, 2759–2774 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals. J. Math. Anal. Appl. 316, 269–306 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Prediction by samples from the past with error estimates covering discontinuous signals. IEEE Trans. Inf. Theory 56(1), 614–633 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bardaro, C., Faina, L., Mantellini, I.: A generalization of the exponential sampling series and its approximation properties. Math. Slovaca 67(6), 1481–1496 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bardaro, C., Mantellini, I.: Approximation properties in abstract modular spaces for a class of general sampling-type operators. Appl. Anal. 85(4), 383–413 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bardaro, C., Mantellini, I.: Asymptotic formulae for linear combinations of generalized sampling type operators. Z. Anal. Anwend. 32(3), 279–298 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Bardaro, C., Musielak, J., Vinti, G.: Nonlinear Integral Operators and Applications. De Gruyter Series in Nonlinear Analysis and Applications, vol. 9. Walter De Gruyter, Berlin (2003)zbMATHGoogle Scholar
  15. 15.
    Bardaro, C., Vinti, G.: An abstract approach to sampling-type operators inspired by the work of P.L. Butzer: I. Linear operators. Sampl. Theory Signal Image Process. 2(3), 271–295 (2003)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Bardaro, C., Vinti, G.: An abstract approach to sampling-type operators inspired by the work of P.L. Butzer: II. Nonlinear operators. Sampl. Theory Signal Image Process. 3(1), 29–44 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Bertero, M., Pike, E.R.: Exponential sampling method for Laplace and other dilationally invariant transforms: I. Singular-system analysis, II. Examples in photon correction spectroscopy and Frauenhofer diffraction. Inverse Probl. 7(1–20), 21–41 (1991)zbMATHGoogle Scholar
  18. 18.
    Butzer, P.L., Dodson, M., Ferreira, P.J.S.G., Higgins, R.J., Schmeisser, G., Stens, R.L.: Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections. Bull. Math. Sci. 4(3), 481–525 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Butzer, P.L., Ferreira, P.J.S.G., Higgins, R.J., Schmeisser, G., Stens, R.L.: The sampling theorem, Poisson’s summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals–their interconnections. Appl. Anal. 90(3–4), 431–461 (2011)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Butzer, P.L., Jansche, S.: A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3, 325–375 (1997)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Butzer, P.L., Jansche, S.: Mellin transform theory and its role of its differential and integral operators. In: Transform Methods and Special Functions, Varna ’96, Bulgarian Academy of Science Sofia, pp. 63–83 (1998)Google Scholar
  22. 22.
    Butzer, P.L., Jansche, S.: The finite Mellin transform, Mellin–Fourier series and the Mellin–Posson summation formula. R. C. Mater. Palermo, Ser. II Suppl. 52, 55–81 (1998)zbMATHGoogle Scholar
  23. 23.
    Butzer, P.L., Jansche, S.: The exponential sampling theorem of signal analysis. Atti Sem. Mat. Fis. Univ. Modena 46, 99–122 (1998)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Butzer, P.L., Jansche, S.: A self-contained approach to Mellin transform analysis for square integrable functions; applications. Integral Transforms Spec. Funct. 8(3–4), 175–198 (1999)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Butzer, P.L., Schmeisser, G., Stens, R.L.: An introduction to sampling analysis. In: Marvasti, F. (ed.) Nonuniform Sampling, Theory and Practice, pp. 17–121. Kluwer, New York (2001)Google Scholar
  26. 26.
    Butzer, P.L., Schmeisser, G., Stens, R.L.: Shannon’s sampling theorem for bandlimited signals and their Hilbert transform, Boas-type formulae for higher order derivatives—the aliasing error involved by their extensions from bandlimited to non-bandlimited signals. Entropy 14, 2192–2226 (2012)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Butzer, P.L., Schmeisser, G., Stens, R.L.: Basic relations valid for the Bernstein space \(B^p_\sigma \) and their extensions to functions from larger spaces with error estimates in terms of their distances from \(B^p_\sigma \). J. Fourier Anal. Appl. 19, 333–375 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Butzer, P.L., Splettstößer, W., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Deutsch. Math. Ver. 90, 1–70 (1988)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Butzer, P.L., Stens, R.L.: Prediction of non-bandlimited signals in terms of splines of low degree. Math. Nachr. 132, 115–130 (1987)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Butzer, P.L., Stens, R.L.: Linear prediction by samples from the past. In: Marks II, R.J. (ed.) Advanced Topics in Shannon Sampling and Interpolation Theory, pp. 157–183. Springer, New York (1993)Google Scholar
  31. 31.
    Casasent, D.: Optical signal processing. In: Casasent, D. (ed.) Optical Data Processing, pp. 241–282. Springer, Berlin (1978)Google Scholar
  32. 32.
    Costarelli, D., Minotti, A.M., Vinti, G.: Approximation of discontinuous signals by sampling Kantorovich series. J. Math. Anal. Appl. 450(2), 1083–1103 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Costarelli, D., Vinti, G.: An inverse result of approximation by sampling Kantorovich series. Proc. Edinb. Math. Soc. 62(1), 265–280 (2019)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Dunford, N., Schwartz, J.T.: Linear Operators, Part I: General Theory. Interscience Publishers, Inc., New York (1958)zbMATHGoogle Scholar
  35. 35.
    Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Wiley, New York (1984)zbMATHGoogle Scholar
  36. 36.
    Gori, F.: Sampling in optics. In: Marks II, R.J. (ed.) Advanced Topics in Shannon Sampling and Interpolation Theory, pp. 37–83. Springer, New York (1993)Google Scholar
  37. 37.
    Haber, S., Shisha, O.: Improper integrals, simple integrals, and numerical quadrature. J. Approx. Theory 11, 1–15 (1974)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis. Foundations. Oxford Univ. Press, Oxford (1996)zbMATHGoogle Scholar
  39. 39.
    Higgins, J.R., Schmeisser, G., Voss, J.J.: The sampling theorem and several equivalent results in analysis. J. Comput. Anal. Appl. 2(4), 333–371 (2000)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Kivinukk, A., Tamberg, G.: Interpolating generalized Shannon sampling operators, their norms and approximation properties. Sampl. Theory Signal Image Process. 8(1), 77–95 (2009)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Mantellini, I.: Generalized sampling operators in modular spaces. Comment. Math. 38, 77–92 (1998)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Mantellini, I., Vinti, G.: Approximation results for nonlinear integral operators in modular spaces and applications. Ann. Polon. Math. 81(1), 55–71 (2003)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Osgood, C.F.: Obtaining a function of bounded coarse variation by a chabge of variable. J. Approx. Theory 44(1), 14–20 (1985)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Osgood, C.F., Shisha, O.: On simple integrability and bounded coarse variation. Approximation theory II. In: Proceedings of International Symposium on University of Texas, Austin, Academic Press, NY, pp. 491–501 (1976)Google Scholar
  45. 45.
    Ostrowsky, N., Sornette, D., Parker, P., Pike, E.R.: Exponential sampling method for light scattering polydispersity analysis. Opt. Acta 28, 1059–1070 (1994)Google Scholar
  46. 46.
    Ries, S., Stens, R.L.: Approximation by generalized sampling series. In: Sendov, B.L., Petrushev, P., Maalev, R., Tashev, S. (eds.) Constructive Theory of Functions, pp. 746–756. Pugl. House Bulgarian Academy of Sciences, Sofia (1984)Google Scholar
  47. 47.
    Schmeisser, G.: Quadrature over a semi-infinite interval and Mellin transform. In: Lyubarskii, Y. (ed.) Proceedings of the 1999 International Workshop on Sampling Theory and Applications, pp. 203–208. Norwegian University of Science and Technology, Trondheim (1999)Google Scholar
  48. 48.
    Schmeisser, G.: Interconnections between the multiplier methods and the window methods in generalized sampling. Sampl. Theory Signal Image Process. 9(1), 1–24 (2010)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Schmeisser, G., Stenger, F.: Sinc approximation with a Gaussian multiplier. Sampl. Theory Signal Image Process. 6(2), 199–221 (2007)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Vinti, G.: A general approximation result for nonlinear integral operators and applications to signal processing. Appl. Anal. 79(1–2), 217–238 (2001)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Zayed, A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer SciencesUniversity of PerugiaPerugiaItaly
  2. 2.Department MathematikFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

Personalised recommendations