Integral Equations and Operator Theory

, Volume 68, Issue 2, pp 193–205 | Cite as

Multipliers for p-Bessel Sequences in Banach Spaces

  • Asghar Rahimi
  • Peter Balazs


Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters.

Mathematics Subject Classification (2010)

Primary 42C15 Secondary 41A58 47A58 


p-Bessel sequence p-Riesz sequence (p, q)-Bessel multiplier (r, p, q)-nuclear operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abodllahpour M.R., Faroughi M.H., Rahimi A.: pg-Frames in Banach spaces. Methods Funct. Anal. Toplol. 13(3), 201–210 (2007)Google Scholar
  2. 2.
    Aldroubi A., Gröchenig K.: Non-uniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 585–620 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aldroubi A., Sun Q., Tang W.-S.: p-Frames and shift invariant subspaces of L p. J. Fourier Anal. Appl. 7(1), 1–21 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Balazs P.: Basic definition and properties of Bessel multipliers. J. Math. Anal. Appl. 325(1), 571–585 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Balazs P.: Hilbert–Schmidt operators and frames—classification, best approximation by multipliers and algorithms. Int. J. Wavelets Multiresolut. Inf. Processing 6(2), 315–330 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Balazs P.: Matrix-representation of operators using frames. Sampling Theory Signal Image Processing (STSIP) 7(1), 39–54 (2008)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Balazs P., Antoine J.-P., Grybos A.: Weighted and controlled frames: mutual relationship and first numerical properties. Int. J. Wavelets Multiresolut. Inf. Processing 8(1), 109–132 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Balazs, P., Deutsch, W.A., Noll, A., Rennison, J., White, J.: STx Programmer Guide, Version: 3.6.2. Acoustics Research Institute, Austrian Academy of Sciences (2005)Google Scholar
  9. 9.
    Balazs P., Laback B., Eckel G., Deutsch W.A.: Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking. IEEE Trans. Audio Speech Lang. Processing 18(1), 34–49 (2010)CrossRefGoogle Scholar
  10. 10.
    Casazza P.G., Christensen O.: Perturbation of operators and applications to frame theory. J. Fourier Anal. Appl. 3(5), 543–557 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Christensen O., Stoeva D.: p-Frames in separable Banach spaces. Adv. Comput. Math. 18(2-4), 117–126 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Depalle, Ph., Kronland-Martinet, R., Torrésani, B.: Time-frequency multipliers for sound synthesis. In: Proceedings of the Wavelet XII conference, SPIE annual Symposium, San Diego, August 2007Google Scholar
  13. 13.
    Dörfler M., Torrésani B.: Representation of operators in the time-frequency domain and generalized gabor multipliers. J. Fourier Anal. Appl. 16(2), 261–293 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Feichtinger, H.G., Nowak, K.: A First Survey of Gabor Multipliers, chap. 5, pp. 99–128. Birkhäuser Boston (2003)Google Scholar
  15. 15.
    Fornasier M., Gröchenig K.: Intrinsic localization of frames. Constr. Approx. 22, 395–415 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Gröchenig K., Heil C.: Modulation spaces and pseudodifferential operators. Integral Equ. Oper. Theory 34(4), 439–457 (1999)CrossRefGoogle Scholar
  17. 17.
    Gröchenig, K., Rzeszotnik, Z.: Banach algebras of pseudodifferential operators and their almost diagonalization. Ann. Inst. Fourier (Grenoble), pp. 2279–2314 (2008)Google Scholar
  18. 18.
    Grothendieck, A.: Produits tensoriels topologiques et espace nucléaires. Mem. Am. Math. Soc. 16 (1955)Google Scholar
  19. 19.
    Kowalski M., Torrésani B.: Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients. Signal Image Video Processing 3(3), 251–264 (2009). doi: 10.1007/s11760-008-0076-1 zbMATHCrossRefGoogle Scholar
  20. 20.
    Matz G., Hlawatsch F.: Linear time-frequency filters: On-line Algorithms and applications. In: Papandreou-Suppappola, A. (eds) Application in Time-Frequency Signal Processing, chap. 6, pp. 205–271. CRC Press, Boca Raton (2002)Google Scholar
  21. 21.
    Matz G., Schafhuber D., Gröchenig K., Hartmann M., Hlawatsch F.: Analysis, optimization, and implementation of low-interference wireless multicarrier systems. IEEE Trans. Wireless Commun. 6(4), 1–11 (2007)CrossRefGoogle Scholar
  22. 22.
    Palmer, T.: Banach Algebras and the General Theory of *-Algebras. Algebras and Banach Algebras (Encyclopedia of Mathematics and its Applications), vol. 1 (1995)Google Scholar
  23. 23.
    Pietsch A.: Operator Ideals. North-Holland, Amsterdam (1980)zbMATHGoogle Scholar
  24. 24.
    Rahimi, A.: Multipliers of generalized frames. Bull. Iranian Math. Soc. (2010, in press)Google Scholar
  25. 25.
    Rauhut H., Schnass K., Vandergheynst P.: Compressed sensing and redundant dictionaries. IEEE Trans. Inform. Theory 54(5), 2210–2219 (2008)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Ruston A.F.: Direct products of banach spaces and linear functional equations. Proc. Lond. Math. Soc. (3) 1, 327–348 (1951)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ruston, A.F.: On the fredholm theory of integral equations for operators belonging to the trace class of general banach spaces. Proc. Lond. Math. Soc, (2), pp. 109–124 (1951)Google Scholar
  28. 28.
    Schatten R.: Norm Ideals of Completely Continuous Operators. Springer, Berlin (1960)zbMATHGoogle Scholar
  29. 29.
    Stoeva, D.: Perturbation of Frames in Banach spaces (submitted)Google Scholar
  30. 30.
    Stoeva, D., Balazs, P.: Unconditional Convergence and Invertibility of Multipliers (2010, submitted)Google Scholar
  31. 31.
    Wang D., Brown G.: Computational Auditory Scene Analysis: Principles, Algorithms, and Applications. Wiley-IEEE Press, New York (2006)Google Scholar
  32. 32.
    Young R.M.: An Introduction to Nonharmonic Fourier Series. Acedmic Press, London (1980)zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MaraghehMaraghehIran
  2. 2.Acoustics Research Institute, Austrian Academy of SciencesViennaAustria

Personalised recommendations