Journal of Fourier Analysis and Applications

, Volume 24, Issue 1, pp 285–308 | Cite as

System Representations for the Paley–Wiener Space \(\mathcal {PW}_{\pi }^2\)

  • Holger Boche
  • Ullrich J. MönichEmail author


In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.


Paley–Wiener space Bandlimited function Linear time-invariant system Convolution sum Divergence \(L^2\)-norm 

Mathematics Subject Classification

94A12 94A20 15A03 



This work was supported by the Gottfried Wilhelm Leibniz Programme of the German Research Foundation (DFG) under Grant BO 1734/20-1, and U. Mönich was supported by the German Research Foundation (DFG) under Grant MO 2572/2-1. It was motivated by questions raised by Hans Feichtinger about sampling-type system approximation processes and the structure of signal sets creating divergence. Parts of this work were presented at the Workshop on Harmonic Analysis, Graphs and Learning at the Hausdorff Research Institute for Mathematics, Bonn, Germany. Holger Boche would like to thank Hans Feichtinger for fruitful discussions and the Hausdorff Research Institute for Mathematics for its support and hospitality.


  1. 1.
    Aron, R.M., Bernal-González, L., Pellegrino, D.M., Seoane-Sepúlveda, J.B.: Lineability: The Search for Linearity in Mathematics. Monographs and Research Notes in Mathematics, vol. 14. CRC Press, Boca Raton (2015)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. 51(1), 71–130 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boche, H., Farrell, B.: Strong divergence of reconstruction procedures for the Paley-Wiener space \({PW}_\pi ^1\) and the Hardy space \({H}^1\). J. Approx. Theory 183, 98–117 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boche, H., Mönich, U.J.: Banach–Steinhaus theory revisited: lineability and spaceability. J. Approx. Theory 213, 50–69 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boche, H., Mönich, U.J.: Signal and system spaces with non-convergent sampling representation. In: Proceedings of European Signal Processing Conference (EUSIPCO), pp. 2131–2135 (2016)Google Scholar
  6. 6.
    Boche, H., Mönich, U.J., Tampubolon, E.: Strong divergence of the Shannon sampling series for an infinite dimensional signal space. In: Proceedings of the 2016 IEEE International Symposium on Information Theory, pp. 2878–2882 (2016)Google Scholar
  7. 7.
    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)MathSciNetzbMATHGoogle Scholar
  8. 8.
    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 derivative–the aliasing error involved by their extensions from bandlimited to non-bandlimited signals. Entropy 14(11), 2192–2226 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Diestel, J.: Sequences and Series in Banach Spaces, vol. 92. Springer, Berlin (2012)Google Scholar
  10. 10.
    Fonf, V.P., Gurariy, V.I., Kadets, M.I.: An infinite dimensional subspace of \(C[0,1]\) consisting of nowhere differentiable functions. C. R. Acad. Bulg. Sci. 52(11–12), 13–16 (1999)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gurariy, V.I.: Subspaces and bases in spaces of continuous functions. Dokl. Akad. Nauk SSSR 167, 971–973 (1966)MathSciNetGoogle Scholar
  12. 12.
    Habib, M.K.: Digital representations of operators on band-limited random signals. IEEE Trans. Inf. Theory 47(1), 173–177 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Higgins, J.R.: Five short stories about the cardinal series. Bull. Am. Math. Soc. 12(1), 45–89 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press, Oxford (1996)zbMATHGoogle Scholar
  15. 15.
    Jerri, A.J.: The Shannon sampling theorem-its various extensions and applications: a tutorial review. Proc. IEEE 65(11), 1565–1596 (1977)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kramer, H.P.: The digital form of operators on band-limited functions. J. Math. Anal. Appl. 44(2), 275–287 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Marvasti, E. (ed.): Nonuniform Sampling: Theory and Practice. Kluwer Academic, Dordrecht (2001)zbMATHGoogle Scholar
  18. 18.
    Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing, 3rd edn. Prentice Hall, Upper Saddle River (2009)zbMATHGoogle Scholar
  19. 19.
    Papoulis, A.: Generalized sampling expansion. IEEE Trans. Circ. Syst. 24(11), 652–654 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37(1), 10–21 (1949)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Stens, R.L.: A unified approach to sampling theorems for derivatives and Hilbert transforms. Signal Process. 5, 139–151 (1983)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zygmund, A.: Trigonometric Series, vol. I, 3rd edn. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Lehrstuhl für Theoretische InformationstechnikTechnische Universität MünchenMunichGermany

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