Abstract
In this paper the convergence behavior of the Shannon sampling series is analyzed for Hardy spaces. It is well known that the Shannon sampling series is locally uniformly convergent. However, for practical applications the global uniform convergence is important. It is shown that there are functions in the Hardy space such that the Shannon sampling series is not uniformly convergent on the whole real axis. In fact, there exists a function in this space such that the peak value of the Shannon sampling series diverges unboundedly. The proof uses Fefferman’s theorem, which states that the dual space of the Hardy space is the space of functions of bounded mean oscillation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beaty, M., Dodson, M.: The Whittaker–Kotel’nikov–Shannon theorem, spectral translates and Plancherel’s formula. J. Fourier Anal. Appl. 10(2), 179–199 (2004)
Boche, H., Mönich, U.J.: On stable Shannon type reconstruction processes. Signal Process. 88(6), 1477–1484 (2008)
Boche, H., Schreiber, H.: The behaviour of the finite Shannon sampling series. In: Proceedings of the 1997 Workshop on Sampling Theory and Applications, SampTA-97, June 1997, pp. 419–424 (1997)
Brown, G., Dai, F., Móricz, F.: The maximal Riesz, Fejér, and Cesàro operators on real Hardy spaces. J. Fourier Anal. Appl. 10(1), 27–50 (2004)
Brown, J.L. Jr.: On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem. J. Math Anal. Appl. 18, 75–84 (1967). Erratum, ibid, 21, 699 (1968)
Butzer, P.L., Splettstößer, W., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Dtsch. Math.-Ver. 90(1), 1–70 (1988)
Butzer, P.L., Stens, R.L.: Sampling theory for not necessarily band-limited functions: a historical overview. SIAM Rev. 34(1), 40–53 (1992)
Campbell, L.L.: Sampling theorem for the Fourier transform of a distribution with bounded support. SIAM J. Appl. Math. 16(3), 626–636 (1968)
García, A., Hernández-Medina, M., Portal, A.: An estimation of the truncation error for the two-channel sampling formulas. J. Fourier Anal. Appl. 11(2), 203–213 (2005)
Garnett, J.B.: In: Eilenberg, S., Bass, H. (eds.) Bounded Analytic Functions. Academic Press, San Diego (1981)
Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis—Foundations. Oxford University Press, London (1996)
Jerri, A.J.: The Shannon sampling theorem–its various extensions and applications: a tutorial review. Proc. IEEE 65(11), 1565–1596 (1977)
Miyachi, A.: Boundedness of the Cesàro operator in Hardy spaces. J. Fourier Anal. Appl. 10(1), 83–92 (2004)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)
Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37(1), 10–21 (1949)
Shannon, C.E.: Communication in the presence of noise. Proc. IEEE 72(9), 1192–1201 (1984)
Unser, M.: Sampling—50 years after Shannon. Proc. IEEE 88(4), 569–587 (2000)
Zygmund, A.: Trigonometric Series, 2nd edn. Cambridge University Press, Cambridge (1993). Vols. I and II
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hans G. Feichtinger.
This work was partly supported by the German Research Foundation (DFG) under grant BO 1734/9-1.
Rights and permissions
About this article
Cite this article
Boche, H., Mönich, U.J. Behavior of Shannon’s Sampling Series for Hardy Spaces. J Fourier Anal Appl 15, 404–412 (2009). https://doi.org/10.1007/s00041-008-9033-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-008-9033-1