Modulation Spaces

  • Árpád Bényi
  • Kasso A. Okoudjou
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


In this chapter, various equivalent definitions of modulation spaces are presented along with their fundamental properties.


  1. 1.
    Balan, R., Christensen, J., Krishtal, I., Okoudjou, K.A., Romero, J.-L.: Multi-window Gabor frames in amalgam spaces. Math. Res. Lett. 21(1), 1–15 (2014)MathSciNetzbMATHGoogle Scholar
  2. 6.
    Bennett, C., Sharpley, R.: Interpolation of operators. In: Pure and Applied Mathematics, vol. 129. Academic, Boston (1988)Google Scholar
  3. 8.
    Bényi, Á., Oh, T.: Modulation spaces, Wiener amalgam spaces, and Brownian motions. Adv. Math. 228(5), 2943–2981 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 26.
    Bertrandias, J.-P., Datry, C., Dupuis, C.: Unions et intersections d’espaces L p invariantes par translation ou convolution. Ann. Inst. Fourier (Grenoble) 28(2), 53–84 (1978)Google Scholar
  5. 27.
    Beurling, A., Helson, H.: Fourier-Stieltjes transform with bounded powers. Math. Scand. 1, 12–126 (1953)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 35.
    Boulkhemair, A.: Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators. Math. Res. Lett. 4, 53–67 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 53.
    Christensen, O.: An introduction to frames and Riesz bases. In: Applied and Numerical Harmonic Analysis. Birkäuser, Boston (2003)Google Scholar
  8. 75.
    Cordero, E., Feichtinger, H., Luef, F.: Banach Gelfand triples for Gabor analysis. In: Pseudo-Differential Operators. Lecture Notes in Mathematics, vol. 1949, pp. 1–33. Springer, Berlin (2008)Google Scholar
  9. 87.
    Daubechies, I.: Time-frequency localization operators: a geometric phase approach. IEEE Trans. Inform. Theory 34(4), 605–612 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 89.
    Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27(5), 1271–1283 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 93.
    Feichtinger, H.G.: On a new Segal algebra. Monatsh. Math. 92(4), 269–289 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 94.
    Feichtinger, H.G.: Modulation spaces on locally compact abelian groups. In: Radha, R., Krishna, M., Thangavelu, S. (eds.) Proceeding of International Conference on Wavelets and Applications (Chennai, 2002), pp. 1–56. New Delhi Allied, New Delhi (2003). Technical report, University of Vienna (1983)Google Scholar
  13. 95.
    Feichtinger, H.G.: Generalized amalgams, with applications to Fourier transform. Can. J. Math. 42(3), 395–409 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 96.
    Feichtinger, H.G.: Modulation spaces: looking back and ahead. Sampl. Theory Signal Image Process. 5(2), 109–140 (2006)MathSciNetzbMATHGoogle Scholar
  15. 97.
    Feichtinger, H.G.: Choosing function spaces in harmonic analysis. In: Excursions in Harmonic Analysis. Applied and Numerical Harmonic Analysis, vol. 4, pp. 65–101. Birkhäuser/Springer, Cham (2015)CrossRefGoogle Scholar
  16. 98.
    Feichtinger, H.G.: Thoughts on numerical and conceptual harmonic analysis. In: New Trends in Applied Harmonic Analysis. Applied and Numerical Harmonic Analysis, pp. 301–329. Birkhäuser/Springer, Berlin (2016)CrossRefGoogle Scholar
  17. 99.
    Feichtinger, H.G., Gröchenig, K.: A unified approach to atomic decompositions via integrable group representations. In: Function Spaces and Applications (Lund, 1986). Lecture Notes in Mathematics, vol. 1302, pp. 52–73. Springer, Berlin (1988)CrossRefGoogle Scholar
  18. 100.
    Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal. 86(2), 307–340 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 101.
    Feichtinger, H.G., Gröchenig, K.: Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146(2), 464–495 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 103.
    Feichtinger, H.G., Strohmer, T.: Gabor analysis and algorithms. In: Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (1998)Google Scholar
  21. 104.
    Feichtinger, H.G., Strohmer, T.: Advances in Gabor analysis. In: Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2003)zbMATHCrossRefGoogle Scholar
  22. 105.
    Feichtinger, H.G., Zimmermann, G.: A Banach space of test functions for Gabor analysis. In: Gabor Analysis and Algorithms, pp. 123–170. Birkhäuser, Boston (1998)zbMATHCrossRefGoogle Scholar
  23. 111.
    Gabor, D.: Theory of communication. J. IEE Lond. 93, 429–457 (1946)Google Scholar
  24. 119.
    Grafakos, L.: Classical Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, vol. 249. Springer, New York (2008)Google Scholar
  25. 120.
    Grafakos, L.: Modern Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, vol. 250. Springer, New York (2009)zbMATHCrossRefGoogle Scholar
  26. 125.
    Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)zbMATHCrossRefGoogle Scholar
  27. 128.
    Gröchenig, K., Heil, C.: Gabor meets Littlewood-Paley: Gabor expansions in \(L^{p}(\mathbb {R}^{d})\). Studia Math. 146, 115–33 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 130.
    Gröchenig, K., Leinert, M.: Wiener’s lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17(1), 1–18 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 132.
    Gröchenig, K., Heil, C., Okoudjou, K.A.: Gabor analysis in weighted amalgam spaces. Sampl. Theory Signal Image Process. 1(3), 225–259 (2002)MathSciNetzbMATHGoogle Scholar
  30. 136.
    Guo, B., Wang, B.X., Zhao, L.: Isometric decomposition operators, function spaces \(E^\lambda _{p, q}\) and applications to nonlinear evolution equations. J. Funct. Anal. 233, 1–39 (2006)Google Scholar
  31. 145.
    Heil, C., Walnut, D.: Continuous and discrete wavelet transforms. SIAM Rev. 31(4), 628–666 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 146.
    Heil, C., Ramanathan, J., Topiwala, P.: Linear independence of time-frequency translates. Proc. Am. Math. Soc. 124(9), 2787–2795 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 148.
    Hernández, E., Weiss, G.: A First Course on Wavelets. CRC, Boca Raton (1996)zbMATHCrossRefGoogle Scholar
  34. 157.
    Jakobsen, M.S.: On a (no longer) new Segal algebra—a review of the Feichtinger algebra. J. Fourier Anal. Appl. 24(6), 1579–1660 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 166.
    Kobayashi, M.: Modulation spaces M p, q for 0 < p, q ≤. J. Funct. Spaces Appl. 4(3), 329–341 (2006)Google Scholar
  36. 173.
    Krishtal, I., Okoudjou, K.A.: Invertibility of the Gabor frame operator on the Wiener amalgam space. J. Approx. Theory 153(2), 212–214 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 176.
    Lebedev, V., Olevskiı̌, A.: C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers. Geom. Funct. Anal. 4(2), 213–235 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 194.
    Okoudjou, K.A.: A Beurling-Helson type theorem for modulation spaces. J. Funct. Spaces Appl. 7(1), 33–41 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 195.
    Okoudjou, K.A.: An invitation to Gabor analysis. Notices Am. Math. Soc. 66(6), 808–819 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 198.
    Rauhut, H.: Coorbit space theory for quasi-Banach spaces. Studia Math. 180(3), 237–253 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 199.
    Ron, A., Shen, Z.: Weyl-Heisenberg frames and Riesz bases in \(L_2(\mathbb {R}^d)\). Duke Math. J. 89(2), 237–282 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 202.
    Runst, T., Sickel, W.: Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. In: de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter, Berlin (1996)Google Scholar
  43. 203.
    Ruzhansky, M., Sugimoto, M., Toft, J., Tomita, N.: Changes of variables in modulation and Wiener amalgam spaces. Math. Nachr. 284(16), 2078–2092 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 205.
    Self, W.M.: Some consequences of the Beurling-Helson theorem. Rocky Mountain J. Math. 6, 177–180 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 206.
    Sjöstrand, J.: An algebra of pseudodifferential operators. Math. Res. Lett. 1(2), 185–192 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 211.
    Sugimoto, M., Tomita, N.: The dilation property of modulation spaces and their inclusion relation with Besov spaces. J. Funct. Anal. 248(1), 79–106 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 220.
    Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I. J. Funct. Anal. 207, 399–429 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 230.
    Walnut, D.F.: Continuity properties of the Gabor frame operator. J. Math. Anal. Appl. 165(2), 479–504 (1992)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  • Árpád Bényi
    • 1
  • Kasso A. Okoudjou
    • 2
  1. 1.Department of MathematicsWestern Washington UniversityBellinghamUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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