Problems of Information Transmission

, Volume 53, Issue 2, pp 164–182 | Cite as

Spaceability for sets of bandlimited input functions and stable linear time-invariant systems with finite time blowup behavior

  • H. BocheEmail author
  • U. J. Mönich
Methods of Signal Processing


The approximation of linear time-invariant systems by sampling series is studied for bandlimited input functions in the Paley–Wiener space PW π 1 , i.e., bandlimited signals with absolutely integrable Fourier transform. It has been known that there exist functions and systems such that the approximation process diverges. In this paper we identify a signal set and a system set with divergence, i.e., a finite time blowup of the Shannon sampling expression. We analyze the structure of these sets and prove that they are jointly spaceable, i.e., each of them contains an infinite-dimensional closed subspace such that for any function and system pair from these subspaces, except for the zero elements, we have divergence.


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  1. 1.
    Boche, H. and Mönich, U.J, Signal and System Spaces with Non-Convergent Sampling Representation, in Proc. 24th European Signal Processing Conf. (EUSIPCO’2016), Budapest, Hungary, Aug. 29–Sept. 2, 2016, pp. 2131–2135.Google Scholar
  2. 2.
    Boche, H. and Mönich, U.J, General Behavior of Sampling-Based Signal and System Representation, in Proc. 2008 IEEE Int. Sympos. on Information Theory (ISIT’2008), Toronto, Ontario, Canada, July 6–11, 2008, pp. 2439–2443.CrossRefGoogle Scholar
  3. 3.
    Boche, H. and Mönich, U.J, Sampling of Deterministic Signals and Systems, IEEE Trans. Signal Process., 2011, vol. 59, no. 5, pp. 2101–2111.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boche, H. and Mönich, U.J, No-Go Theorem for Sampling-Based Signal Processing, in Proc. 2014 IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP’14), Florence, Italy, May 4–9, 2014, pp. 56–60.CrossRefGoogle Scholar
  5. 5.
    Fonf, V.P., Gurariy, V.I., and Kadets, M.I, An Infinite Dimensional Subspace of C[0, 1] Consisting of Nowhere Differentiable Functions, C. R. Acad. Bulgare Sci., 1999, vol. 52, no. 11–12, pp. 13–16.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gurariy, V.I. and Quarta, L, On Lineability of Sets of Continuous Functions, J. Math. Anal. Appl., 2004, vol. 294, no. 1, pp. 62–72.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Aron, R., Gurariy, V.I., and Seoane, J.B, Lineability and Spaceability of Sets of Functions on R, Proc. Amer. Math. Soc., 2005, vol. 133, no. 3, pp. 795–803.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bayart, F, Linearity of Sets of Strange Functions, Michigan Math. J., 2005, vol. 53, no. 2, pp. 291–303.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Botelho, G., Diniz, D., Fávaro, V.V., and Pellegrino, D, Spaceability in Banach and Quasi-Banach Sequence Spaces, Linear Algebra Appl., 2011, vol. 434, no. 5, pp. 1255–1260.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bernal-González, L., Pellegrino, D., and Seoane-Sepúlveda, J.B, Linear Subsets of Nonlinear Sets in Topological Vector Spaces, Bull. Amer. Math. Soc. (N.S.), 2014, vol. 51, no. 1, pp. 71–130.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Aron, R.M., Bernal-González, L., Pellegrino, D.M., and Seoane-Sepúlveda, J.B., Lineability: The Search for Linearity in Mathematics, Boca Raton: CRC Press, 2016.zbMATHGoogle Scholar
  12. 12.
    Gurariy, V.I, Subspaces and Bases in Spaces of Continuous Functions, Dokl. Akad. Nauk SSSR, 1966, vol. 167, no. 5, pp. 971–973.MathSciNetGoogle Scholar
  13. 13.
    García-Pacheco, F.J., Palmberg, N., and Seoane-Sepúlveda, J.B, Lineability and Algebrability of Pathological Phenomena in Analysis, J. Math. Anal. Appl., 2007, vol. 326, no. 2, pp. 929–939.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Aron, R.M., García-Pacheco, F.J., Pérez-García, D., and Seoane-Sepúlveda, J.B, On Dense-Lineability of Sets of Functions on R, Topology, 2009, vol. 48, no. 2–4, pp. 149–156.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bernal-González, L. and Ordónez Cabrera, M, Lineability Criteria, with Applications, J. Funct. Anal., 2014, vol. 266, no. 6, pp. 3997–4025.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Boche, H., Mönich, U.J., and Tampubolon, E, Strong Divergence of the Shannon Sampling Series for an Infinite Dimensional Signal Space, Proc. 2016 IEEE Int. Sympos. on Information Theory (ISIT’2016), Barcelona, Spain, July 10–15, 2016, pp. 2878–2882.CrossRefGoogle Scholar
  17. 17.
    Brown, J.L, Bounds for Truncation Error in Sampling Expansions of Band-Limited Signals, IEEE Trans. Inform. Theory, 1969, vol. 15, no. 4, pp. 440–444.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Butzer, P.L., Splettstößer, W., and Stens, R.L, The Sampling Theorem and Linear Prediction in Signal Analysis, Jahresber. Deutsch. Math.-Verein., 1988, vol. 90, no. 1, pp. 1–70.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Zygmund, A., Trigonometric Series, vol. I, Cambridge, UK; New York: Cambridge Univ. Press,2002, 3rd ed.zbMATHGoogle Scholar
  20. 20.
    Duren, P.L., Theory of Hp Spaces, New York: Academic, 1970.zbMATHGoogle Scholar
  21. 21.
    Diestel, J., Sequences and Series in Banach Spaces, New York: Springer-Verlag, 1984.CrossRefzbMATHGoogle Scholar
  22. 22.
    Boche, H. and Mönich, U.J, System Representations for the Paley–Wiener Space PW2 p, accepted for publication in J. Fourier Anal. Appl., 2017, doi:10.1007/s00041-016-9517-3.Google Scholar
  23. 23.
    Banach, S. and Steinhaus, H, Sur le principe de la condensation de singularités, Fundam. Math., 1927, vol. 9, pp. 50–61.zbMATHGoogle Scholar
  24. 24.
    Banach, S., Über die Bairesche Kategorie gewisser Funktionenmengen, Stud. Math., 1931, vol. 3, no. 1, pp. 174–179.zbMATHGoogle Scholar
  25. 25.
    Kantorovich, L.V. and Akilov G.P., Funktsional’nyi analiz v normirovannykh prostranstvakh, Moscow: Fizmatlit, 1959. Translated under the title Functional Analysis in Normed Spaces, Oxford; New York: Pergamon, 1964.zbMATHGoogle Scholar

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© Pleiades Publishing, Inc. 2017

Authors and Affiliations

  1. 1.Technische Universität MünchenLehrstuhl für Theoretische InformationstechnikMunichGermany

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