Skip to main content
Log in

Modulation Spaces: Looking Back and Ahead

  • Published:
Sampling Theory in Signal and Image Processing Aims and scope Submit manuscript

Abstract

This note provides historical perspective and background on the motivations which led to the invention of the modulation spaces by the author almost 25 years ago, as well as comments about their role for ongoing research efforts within time-frequency analysis. We will also describe the role of the modulation spaces within the more general coorbit theory developed jointly with Karlheinz Gröchenig, and which eventually led to the development of the concept of Banach frames and more recently to the so-called localization theory for frames. A comprehensive bibliography is included.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. J. B. Allen and L. R. Rabiner, A unified approach to short-time Fourier analysis and synthesis, Proc. IEEE, 65(11):1558–1564, 1977.

    Article  Google Scholar 

  2. R. Ashino, P. Boggiatto, and M. W. Wong, editors, Advances in Pseudo-Differential Operators, vol. 155 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2004.

    Google Scholar 

  3. R. Ashino, M. Nagase, and R. Vaillancourt, Gabor, wavelet and chirplet transforms in the study of pseudodifferential operators. Investigations on the structure of solutions to partial differential equations (Kyoto, 1997), Sūrikaisekikenkyūsho Kōkyūroku, (1036):23–45, 1998.

    MATH  Google Scholar 

  4. P. Auscher, G. Weiss, and M. V. Wickerhauser, Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets, In C. K. Chui, editor, Wavelets: A Tutorial in Theory and Applications, pages 237–256, Academic Press, Boston, 1992.

    Chapter  MATH  Google Scholar 

  5. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Deficits and excesses of frames, Adv. Comput. Math., 18(2-4):93–116, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Density, overcom-pleteness, and localization of frames, I. Theory, J. Fourier Anal. Appl., 12(2):105–143, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Density, overcomplete-ness, and localization of frames, II. Gabor systems, J. Fourier Anal. Appl., to appear, 2006.

    Google Scholar 

  8. J. J. Benedetto, W. Czaja, and A. M. Powell, An optimal example for the Balian-Low uncertainty principle, SIAM J. Math. Anal., to appear, 2006.

    Google Scholar 

  9. J. J. Benedetto, C. Heil, and D. F. Walnut, Differentiation and the Balian-Low theorem, J. Fourier Anal. Appl., 1(4) 355–402, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. J. Benedetto and G. E. Pfander, Frame expansions for Gabor multipliers, Appl. Comput. Harm. Anal., 20(1):26–40, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  11. Á. Bényi, L. Grafakos, K. Gröchenig, and K. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal., 19(1):131–139, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  12. Á. Bényi, K. Gröchenig, C. Heil, and K. Okoudjou, Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Operator Theory, 54(2):387–399, 2005.

    MathSciNet  MATH  Google Scholar 

  13. Á. Bényi and K. Okoudjou, Bilinear pseudo differential operators on modulation spaces, J. Fourier Anal. Appl., 10(3):301–313, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, vol. 223 of Grundlehren Math. Wiss., Springer, Berlin, 1976.

  15. J. P. Bertrandias, C. Datry, and C. Dupuis, Unions et intersections d'espaces lp invariantes par translation ou convolution, Ann. Inst. Fourier, 28(2):53–84, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. P. Bertrandias and C. Dupuis, Transformation de Fourier sur les espaces lp(Lp'), Ann. Inst. Fourier, 29(1):189–206, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Bittner, Linear approximation and reproduction of polynomials by Wilson bases, J. Fourier Anal. Appl., 8(1):85–108, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  18. K. Bittner and K. Gröchenig, Direct and inverse approximation theorems for local trigonometric bases, J. Approx. Theory, 117(1):74–102, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Boggiatto, E. Cordero, and K. Gröchenig, Generalized anti-Wick operators with symbols in distributional Sobolev spaces, Integral Equations Operator Theory, 48(4):427–442, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Boggiatto and J. Toft, Embeddings and compactness for generalized Sobolev-Shubin spaces and modulation spaces, Appl. Anal., 84(3):269–282, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  21. L. Borup, Pseudodifferential operators on a-modulation spaces, J. Funct. Spaces Appl., 2(2):107–123, {sy2004}.

  22. L. Borup and M. Nielsen, Boundedness for pseudodifferential operators on multivariate a-modulation spaces, Ark. Mat., to appear, 2006.

    Google Scholar 

  23. L. Borup and M. Nielsen, Banach frames for multivariate a-modulation spaces, J. Math. Anal. Appl., to appear, 2006.

    Google Scholar 

  24. L. Borup and M. Nielsen, Nonlinear approximation in a-modulation spaces, Math. Nachr., 279(1-2):101–120, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Bownik and K.-P. H, Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Trans. Amer. Math. Soc., 358(4): 1469–1510, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. C. Busby and H. A. Smith, Product-convolution operators and mixed-norm spaces, Trans. Amer. Math. Soc., 263:309–341, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  27. O. Christensen, Atomic decomposition via projective group representations, Rocky Mt. J. Math., 26(4):1289–1312, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. R. Coifman, G. Matviyenko, and Y. Meyer, Modulated Malvar-Wilson bases, Appl. Comput. Harmon. nal., 4(1):58–61, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  29. R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83(4):569–645, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  30. E. Cordero, Gelfand-Shilov windows for weighted modulation spaces, Preprint, 2006.

    Google Scholar 

  31. E. Cordero and K. Gröochenig, Time-frequency analysis of localization operators, J. Funct. Anal., 205(1):107–131, 2003.

    Article  MathSciNet  Google Scholar 

  32. E. Cordero and K. Gröochenig, Localization of frames II, Appl. Comput. Harmon. Anal., 17(1):29–47, 2004.

  33. E. Cordero and K. Groöchenig, Symbolic calculus and Fredholm property for localization operators, Preprint, 2005.

    Google Scholar 

  34. E. Cordero and L. Rodino, Short-time Fourier transform analysis of localization operators, Preprint, 2006.

    Google Scholar 

  35. E. Cordero, S. Pilipovic, L. Rodino, and N. Teofanov, Localization operators and exponential weights for modulation spaces, Mediterr. J. Math., 2(4):381–394, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Cordero and L. Rodino, Wick calculus: A time-frequency approach, Osaka J. Math., 42(1):43–63, 2005.

    MathSciNet  MATH  Google Scholar 

  37. E. Cordero and A. Tabacco, Localization operators via time-frequency analysis, In Advances in Pseudo-Differential Operators, vol. 155 of Oper. Theory Adv. Appl., pages 131–147, Birkhöauser, Basel, 2004.

    Chapter  MATH  Google Scholar 

  38. W. Czaja, Boundedness of pseudodifferential operators on modulation spaces, J. Math. Anal. Appl., 284(1):389–396, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  39. S. Dahlke, G. Steidl, and G. Teschke, Coorbit spaces and Banach frames on homogeneous spaces with applications to the sphere, Adv. Comput. Math., 21(1-2):147–180, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  40. S. Dahlke, M. Fornasier, H. Rauhut, G. Steidl, and G. Teschke, Generalized coorbit theory, Banach frames, and the relation to alpha-modulation spaces, Preprint, 2005.

    MATH  Google Scholar 

  41. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36(5):961–1005, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  42. I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27(5):1271–1283, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  43. I. Daubechies, S. Jaffard, and J. L. Journe, A simple Wilson orthonormal basis with exponential decay, SIAM J. Math. Anal., 22(2):554–573, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  44. T. Dobler, Wiener Amalgam Spaces on Locally Compact Groups, Master's thesis, University of Vienna, 1989: http://www.mat.un.ivie.ac.at/~nuhag/papers/PS/1989/dob89.zip

    Google Scholar 

  45. D. L. Donoho, Unconditional bases are optimal bases for data compression and for statistical estimation, Appl. Comput. Harmon. Anal., 1(1):100–115, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  46. D. L. Donoho, M. Vetterli, R. A. DeVore, and I. Daubechies, Data compression and harmonic analysis, IEEE Trans. Inform. Theory, 44(6):2435–2476, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Dörfler, H. G. Feichtinger, and K. Grochenig, Compactness criteria in function spaces, Colloq. Math., 94(1):37–50, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  48. M. Duflo and C. C. Moore, On the regular representation of a nonuni-modular locally compact group, J. Funct. Anal., 21:209–243, 1976.

    Article  MATH  Google Scholar 

  49. H. G. Feichtinger, A characterization of Wiener's algebra on locally compact groups, Arch. Math. (Basel), 29(2):136–140, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  50. H. G. Feichtinger, On a class of convolution algebras of functions, Ann. Inst. Fourier (Grenoble), 27(3):135–162, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  51. H. G. Feichtinger, Gewichtsfunktionen auf lokalkompakten Gruppen, Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 188(8-10):451–471, 1979.

    MathSciNet  MATH  Google Scholar 

  52. H. G. Feichtinger, Eine neue Segalalgebra mit Anwendungen in der Harmonischen Analyse, Winterschule 1979, Internationale Arbeitstagung über topologische Gruppen und Gruppenalgebren, Wien, pages 23–25. February 1979.

    Google Scholar 

  53. H. G. Feichtinger, Un espace de Banach de distributions tempérées sur les groupes localement compacts abeliens, C. R. Acad. Sci. Paris Ser. A-B, 290(17):791–794, 1980.

    MathSciNet  MATH  Google Scholar 

  54. H. G. Feichtinger, A characterization of minimal homogeneous Banach spaces, Proc. Amer. Math. Soc., 81(1):55–61, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  55. H. G. Feichtinger, Banach spaces of distributions of Wiener s type and interpolation, In P. Butzer, S. B. Nagy, and E. Göorlich, editors, Proc. Conf. Functional Analysis and Approximation, Oberwolfach August 1980, vol. 60 in Internat. Ser. Numer. Math., pages 153–165, Birkhöauser Boston, Basel, 1981.

    Google Scholar 

  56. H. G. Feichtinger, On a new Segal algebra, Monatsh. Math., 92(4):269–289, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  57. H. G. Feichtinger, Banach spaces of distributions defined by decomposition methods and some of their applications, In Recent Trends in Mathematics, Proc. Conf. Reinhardsbrunn, vol. 50 of Teubner Texte zur Mathematik, pages 123–132, Teubner, 1982.

    Google Scholar 

  58. H. G. Feichtinger, Modulation spaces on locally compact Abelian groups, Technical report, Univ. Vienna, 52 pages, January 1983.

    Google Scholar 

  59. H. G. Feichtinger, Banach convolution algebras of Wiener type, In Proc. Conf. on Functions, Series, Operators, Budapest 1980, vol. 35 of Colloq. Math. Soc. János Bolyai, pages 509–524, North-Holland, Amsterdam, 1983.

    Google Scholar 

  60. H. G. Feichtinger, A new family of functional spaces on the Euclidean n-space, In Proc. Conf. on Theory of Approximation of Functions, Teor. Priblizh., 1983.

    Google Scholar 

  61. H. G. Feichtinger, Compactness in translation invariant Banach spaces of distributions and compact multipliers, J. Math. Anal. Appl., 102(2):289–327, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  62. H. G. Feichtinger, Banach spaces of distributions defined by decomposition methods, II, Math. Nachr., 132:207–237, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  63. H. G. Feichtinger, An elementary approach to the generalized Fourier transform, In T. Rassias, editor, Topics in Mathematical Analysis, pages 246–272, Volume in honor of Cauchy's 200th anniversary, World Sci.Pub., 1989.

    Chapter  Google Scholar 

  64. H. G. Feichtinger, Atomic characterizations of modulation spaces through Gabor-type representations, Rocky Mountain J. Math., 19(1):113–125, 1989. Constructive Function Theory—86 Conference (Edmonton, AB, 1986).

    Article  MathSciNet  MATH  Google Scholar 

  65. H. G. Feichtinger, Generalized amalgams, with applications to Fourier transform, Canad. J. Math., 42(3):395–409, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  66. H. G. Feichtinger, New results on regular and irregular sampling based on Wiener amalgams, In K. Jarosz, editor, Function Spaces (Edwardsville, IL, 1990), vol. 136 of Lect. Notes Pure Appl. Math., pages 107–121, Dekker, New York, 1992.

    Google Scholar 

  67. H. G. Feichtinger. Wiener amalgams over Euclidean spaces and some of their applications, In K. Jarosz, editor, Function Spaces (Edwardville, IL, 1990), vol. 136 of Lecture Notes in Pure and Appl. Math., pages 123–137, Dekker, New York, 1992.

    Google Scholar 

  68. H. G. Feichtinger, Amalgam spaces and generalized harmonic analysis, In V. Mendrekar {etet al.}, editors, Proceedings ofthe Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994) vol. 52 of Proc. Sympos. Appl. Math., pages 141–150, American Mathematical Society, Providence, RI, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  69. H. G. Feichtinger, Spline-type spaces in Gabor analysis, In Wavelet Analysis (Hong Kong, 2001), vol. 1 of Ser. Anal., pages 100–122, World Sci. Publishing, River Edge, NJ, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  70. H. G. Feichtinger, Modulation spaces of locally compact Abelian groups. In M. Krishna, R. Radha, and S. Thangavelu, editors, Wavelets and their Applications (Chennai, January 2002), pages 1–56, Allied Publishers, New Delhi, 2003.

    Google Scholar 

  71. H. G. Feichtinger and M. Fornasier, Flexible Gabor-wavelet atomic decompositions for L2 Sobolev spaces, Ann. Mat. Pura e Appl. (4), 185(1):105–131, 2006.

    Article  MATH  Google Scholar 

  72. H. G. Feichtinger and P. Gröbner. Banach spaces of distributions defined by decomposition methods, I. Math. Nachr., 123:97–120, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  73. H. G. Feichtinger and K. Gröchenig, A unified approach to atomic decompositions via integrable group representations, In Function Spaces and Applications (Lund, 1986), vol. 1302 of Lecture Notes in Math., pages 52–73, Springer, Berlin, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  74. H. G. Feichtinger and K. Gröochenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal., 86(2):307–340, 1989.

    Article  MathSciNet  Google Scholar 

  75. H. G. Feichtinger and K. Gröochenig, Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsh. Math., 108(2-3):129–148, 1989.

    Article  MathSciNet  Google Scholar 

  76. H. G. Feichtinger and K. Gröochenig, Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view, In C. K. Chui, editor, Wavelets, vol. 2 of Wavelet Anal. Appl., pages 359–397, Academic Press, Boston, 1992.

    Chapter  Google Scholar 

  77. H. G. Feichtinger and K. Gröochenig, Gabor frames and time-frequency analysis of distributions, J. Funct. Anal., 146(2):464–495, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  78. H. G. Feichtinger, K. Gröochenig, and D. Walnut, Wilson bases and modulation spaces, Math. Nachr., 155:7–17, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  79. H. G. Feichtinger, M. Hampejs, and G. Kracher, Approximation of matrices by Gabor multipliers, IEEE Signal Proc. Letters, 11(11):883–886, 2004.

    Article  Google Scholar 

  80. H. G. Feichtinger and A. J. E. M. Janssen, Validity of WH-frame bound conditions depends on lattice parameters, Appl. Comput. Harmon. Anal., 8(1):104–112, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  81. H. G. Feichtinger and N. Kaiblinger, Varying the time-frequency lattice of Gabor frames, Trans. Amer. Math. Soc., 356(5):2001–2023, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  82. H. G. Feichtinger and N. Kaiblinger, Quasi-interpolation in the Fourier algebra, J. Approx. Theory, to appear, 2006.

    Google Scholar 

  83. H. G. Feichtinger and W. Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, Chap. 7 in [89], pages 233–266, 1998.

    MATH  Google Scholar 

  84. H. G. Feichtinger and F. Luef, Wiener amalgam spaces for the fundamental identity of Gabor analysis, In Proc. Conf. El-Escorial, 2004, El-Escorial, 2005. Collect. Math. to appear, 2006.

    Google Scholar 

  85. H. G. Feichtinger and G. Narimani, Fourier Multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal., to appear, 2006.

    Google Scholar 

  86. H. G. Feichtinger and K. Nowak, A first survey of Gabor multipliers, Chap. 5 in [90], pages 99–128, 2003.

    MATH  Google Scholar 

  87. H. G. Feichtinger and S. S. Pandey, Error estimates for irregular sampling of band-limited functions on a locally compact Abelian group, J. Math. Anal. Appl., 279(2):380–397, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  88. H. G. Feichtinger and T. Strohmer, editors, Gabor analysis and algorithms. Theory and Applications, Birkhöauser, Boston, 1998.

    MATH  Google Scholar 

  89. H. G. Feichtinger and T. Strohmer, editors, Advances in Gabor Analysis, Birkhaöuser, Boston, 2003.

    MATH  Google Scholar 

  90. H. G. Feichtinger and T. Werther, Robustness of minimal norm interpolation in Sobolev algebras, In J. J. Benedetto and A. Zayed, editors, Sampling, Wavelets, and Tomography, pages 83–113, Birkhöauser, Boston, 2002.

    MATH  Google Scholar 

  91. H. G. Feichtinger and F. Weisz, The Segal algebra S0 (Rd) and norm summability of Fourier series and Fourier transforms, Monatsh. Math., to appear, 2006.

    Google Scholar 

  92. H. G Feichtinger and F. Weisz, Inversion formulas for the short-time Fourier transform, Preprint, 2006.

    Book  MATH  Google Scholar 

  93. H. G. Feichtinger and G. Zimmermann, A Banach space of test functions for Gabor analysis, Chap. 3 in [89], pages 123–170, 1998.

    MATH  Google Scholar 

  94. C. Fernandez and A. Galbis, Compactness of time-frequency localization operators on L2(R), J. Funct. Anal., 233(2):335–350, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  95. G. B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989.

    Book  MATH  Google Scholar 

  96. M. Fornasier, Constructive Methods for Numerical Applications in Signal Processing and Homogenization Problems, Ph.D. thesis, Univ. Padova and Univ. Vienna, 2002.

    Google Scholar 

  97. M. Fornasier, Banach frames for alpha-modulation spaces, Preprint, 2004.

    MATH  Google Scholar 

  98. M. Fornasier and K. Gröochenig, Intrinsic localization of frames, Constr. Approx., 22(3):395–415, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  99. M. Fornasier and H. Rauhut, Continuous frames, function spaces, and the discretization problem, J. Fourier Anal. Appl., 11(3):245–287, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  100. J. J. F. Fournier and J. Stewart, Amalgams of Lp and Lq, Bull. Am. Math. Soc. (N.S.), 13(1):1–21, 1985.

    Article  Google Scholar 

  101. M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34(4):777–799, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  102. M. Frazier and B. Jawerth, The ϕ-transform and applications to distribution spaces, In Function Spaces and Applications (Lund, 1986), vol. 1302 of Lecture Notes in Math., pages 223–246, Springer, Berlin, 1988.

    Chapter  Google Scholar 

  103. M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal., 93(1):34–170, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  104. M. Frazier and B. Jawerth, Applications of the ϕ and wavelet transforms to the theory of function spaces, In M. B. Ruskai {etet al.}, editors, Wavelets and Their Applications, pages 377–417, Jones and Bartlett, Boston, 1992.

    MATH  Google Scholar 

  105. D. Gabor, Theory of communication, J. IEE, 93(26):429–457, 1946.

    Google Scholar 

  106. Y. V. Galperin and K. Groöchenig, Uncertainty principles as embeddings of modulation spaces, J. Math. Anal. Appl., 274(1):181–202, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  107. Y. V. Galperin and S. Samarah, Time-frequency analysis on modulation spaces Mmpq, 0 < p, q ≤ ∞}, Appl. Comput. Harmon. Anal., 16(1):1–18, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  108. J. E. Gilbert and J. D. Lakey, On a characterization of the local Hardy space by Gabor frames, In Wavelets, Frames and Operator Theory, vol. 345 of Contemp. Math., pages 153–161, Amer. Math. Soc., Providence, RI, 2004.

    Chapter  MATH  Google Scholar 

  109. N. Grip and G. E. Pfander, A discrete model for the efficient analysis of time-varying narrowband communication channels, Preprint, 2005.

    MATH  Google Scholar 

  110. P. Gröbner, Banachräume glatter Funktionen und Zerlegungsmethoden, Ph.D. Thesis, University of Vienna, 1992.

    Google Scholar 

  111. K. Gröchenig, Describing functions: Atomic decompositions versus frames, Monatsh. Math., 112(3):1–41, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  112. K. Groöchenig, An uncertainty principle related to the Poisson summation formula, Studia Math., 121(1):87–104, 1996.

    Article  MathSciNet  Google Scholar 

  113. K. Gröochenig, Foundations of Time-Frequency Analysis, Birkhöauser, Boston, 2001.

    Book  Google Scholar 

  114. K. Gröochenig, Time-frequency analysis of Sjöostrand's class, Revista Mat. Iberoam., to appear, 2006.

    Google Scholar 

  115. K. Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., 10(2):105–132, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  116. K. Groöchenig, Weight functions in time-frequency analysis, Preprint, 2006.

    Google Scholar 

  117. K. Gröchenig and C. Heil, Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory, 34(4):439–457, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  118. K. Groöchenig and C. Heil, Counterexamples for boundedness of pseudodifferential operators, Osaka J. Math., 41(3):681–691, 2004.

    MathSciNet  MATH  Google Scholar 

  119. K. Groöchenig, C. Heil, and K. Okoudjou, Gabor analysis in weighted amalgam spaces, Sampl. Theory Signal Image Process., 1(3):225–259, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  120. K. Gröochenig and M. Leinert, Wiener's lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc., 17(1):1–18, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  121. K. Gröochenig and S. Samarah, Nonlinear approximation with local Fourier bases, Constr. Approx., 16(3):317–331, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  122. K. Gröochenig and D. Walnut, A Riesz basis for Bargmann-Fock space related to sampling and interpolation, Ark. Mat., 30(2):283–295, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  123. K. Gröchenig and T. Strohmer, Analysis of pseudodifferential operators of Sjöostrand's class on locally compact abelian groups, Preprint, 2006.

    Google Scholar 

  124. K. Grochenig and G. Zimmermann, Spaces of test functions via the STFT, J. Funct. Spaces Appl., 2(1):25–53, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  125. A. Grossmann and J. Morlet, Decomposition of functions into wavelets of constant shape, and related transforms, In Mathematics and Physics, Lect. Recent Results, Bielefeld/FRG 1983/84, vol. 1 of Irreducible Unitary Linear Representations; Connected Lie Group; Kirillov Coadjoint Orbit; Quantization; Discrete Series; Affine Wavelets, pages 135–165, 1985.

    Google Scholar 

  126. M. M. Hartmann, G. Matz, and D. Schafhuber, Theory and design of multipulse multicarrier systems for wireless communications, In Signals, Systems and Computers, 2003. Conference Record of the Thirty-Seventh Asilomar Conference on, Vol. 1, pages 492–496, 2003.

    Google Scholar 

  127. C. Heil, Integral operators, pseudodifferential operators, and Gabor frames, Chap. 7 in [90], pages 153–169, 2003.

    MATH  Google Scholar 

  128. C. Heil, An introduction to weighted Wiener amalgams, In M. Krishna, R. Radha, and S. Thangavelu, editors, Wavelets and their Applications (Chennai, January 2002), pages 183–216, Allied Publishers, New Delhi, 2003.

    Google Scholar 

  129. C. Heil, J. Ramanathan, and P. Topiwala, Singular values of compact pseudodifferential operators, J. Funct. Anal., 150(2):426–452, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  130. C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev., 31(4):628–666, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  131. C. Herz, Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, J. Math. Mech., 18:283–323, 1968/69.

    MathSciNet  MATH  Google Scholar 

  132. J. A. Hogan and J. D. Lakey, Extensions of the Heisenberg group by dilations and frames, Appl. Comput. Harmon. Anal., 2(2):174–199, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  133. J. A. Hogan and J. D. Lakey, Embeddings and uncertainty principles for generalized modulation spaces, In J. J. Benedetto and P. J. S. G. Ferreira, editors, Modern Sampling Theory, pages 73–105, Birkhöauser, Boston, 2001.

    Chapter  Google Scholar 

  134. J. A. Hogan and J. D. Lakey, Time-Frequency and Time-Scale Methods, Birkhaöuser, Boston, 2005.

    MATH  Google Scholar 

  135. W. Höormann, Generalized Stochastic Processes and Wigner Distribution, Ph.D. Thesis, University of Vienna, 1989.

    Google Scholar 

  136. R. Howe, On the role of the Heisenberg group in harmonic analysis, Bull. Am. Math. Soc. (N.S.), 3(2):821–843, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  137. A. J. E. M. Janssen, From continuous to discrete Weyl-Heisenberg frames through sampling, J. Fourier Anal. Appl., 3(5):583–596, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  138. R. Johnson, Lipschitz spaces, Littlewood-Paley spaces, and convoluteurs, Proc. Lond. Math. Soc. (3), 29:127–141, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  139. H. Junek and T. V. Vuong, On modulation spaces, Wiss. Z. Paädagog. Hochsch. “Karl Liebknecht” Potsdam, 32(1):153–162, 1988.

    MathSciNet  MATH  Google Scholar 

  140. J.-P. Kahane and P.-G. Lemarié-Rieusset, Remarques sur la formule som-matoire de Poisson, Studia Math., 109(3):303–316, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  141. N. Kaiblinger, Approximation of the Fourier transform and the dual Gabor window, J. Fourier Anal. Appl., 11(1):25–42, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  142. Y. Katznelson, An Introduction to Harmonic Analysis, Third Edition, Cambridge University Press, Cambridge, 2003.

    MATH  Google Scholar 

  143. B. Keville, Multidimensional Second Order Generalised Stochastic Processes on Locally Compact Abelian Groups, Ph.D. Thesis, Trinity College Dublin, 2003.

    Google Scholar 

  144. W. Kozek, Matched Weyl-Heisenberg Expansions ofNonstationary Environments, Ph.D. Thesis, ienna University of Technology, 1997.

    Google Scholar 

  145. W. Kozek, G. E. Pfander, and G. Zimmermann, Perturbation stability of various coherent Riesz families, In A. Aldroubi, A. F. Laine, and M. A. Unser, editors, Wavelet Applications in Signal and Image Processing VIII (San Diego, CA, 2000) pages 411–419, SPIE, Bellingham, WA, 2000.

    Chapter  Google Scholar 

  146. W. Kozek and G. E. Pfander, Identification of operators with bandlimited symbols, SIAM J. Math. Anal., 37(3):867–888, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  147. M. Krishna, R. Radha, and S. Thangavelu (editors), Wavelets and their Applications (Chennai, January 2002), Allied Publishers, New Delhi, 2003.

    Google Scholar 

  148. G. Kutyniok and T. Strohmer, Wilson bases for general time-frequency lattices, SIAM J. Math. Anal. 37(3), 685–711, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  149. D. Labate, Time-frequency analysis of pseudodifferential operators, Monatsh. Math., 133(2):143–156, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  150. D. Labate, Pseudodifferential operators on modulation spaces, J. Math. Anal. Appl., 262(1):242–255, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  151. N. Lerner, On the Fefferman-Phong inequality and a Wiener-type algebra of pseudodifferential operators, Preprint, 2006.

    MATH  Google Scholar 

  152. V. Losert, A characterization of the minimal strongly character invariant Segal algebra, Ann. Inst. Fourier (Grenoble), 30(3):129–139, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  153. V. Losert, Segal algebras with functional poperties, Monatsh. Math., 96(3):209–231, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  154. F. Luef, Gabor Analysis meets Noncommutative Geometry, Ph.D. Thesis, University of Vienna, 2005.

  155. F. Luef, Gabor analysis, noncommutative tori and Feichtinger's algebra, Preprint, 2006.

    MATH  Google Scholar 

  156. G. Matz, D. Schafhuber, K. Gröochenig, M. Hartmann, and F. Hlawatsch, Analysis, optimization, and implementation of low-interference wireless multicarrier systems, Preprint, 2005.

    Google Scholar 

  157. Y. Meyer, De la recherche petrolière à la géometrie des espaces de Banach en passant par les paraproduite, 1986, In Sminaire sur les equations aux dérivé partielles, 1985-1986, Exp. No. I, École Polytech., Palaiseau, 1986.

    Google Scholar 

  158. K. A. Okoudjou, Embedding of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc., 132(6):1639–1647, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  159. L. Päivärinta and E. Somersalo, A generalization of the Calderón-Vaillancourt theorem to lp and hp, Math. Nachr., 138:145–156, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  160. S. S. Pandey, Wavelet representation of modulated spaces on locally compact abelian groups, Ganita, 50(2):119–128, 1999.

    MathSciNet  MATH  Google Scholar 

  161. S. S. Pandey, Time-frequency localizations for modulation spaces on locally compact abelian groups, Int. J. Wavelets Multiresolut. Inf. Process., 2(2):149–163, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  162. A. Papoulis, Signal Analysis, McGraw-Hill Book Company, New York, 1977.

    MATH  Google Scholar 

  163. J. Peetre, New Thoughts on Besov Spaces, Mathematics Department, Duke University, Durham, NC, 1976.

    MATH  Google Scholar 

  164. G. E. Pfander and D. Walnut, Measurement of time-varying channels, Preprint, 2005.

    MATH  Google Scholar 

  165. G. E. Pfander and D. Walnut, Operator identification and Feichtinger's algebra, Sampl. Theory Signal Image Process., 5(2), 2006.

    Google Scholar 

  166. S. Pilipovic and N. Teofanov, Wilson bases and ultramodulation spaces. Math. Nachr., 242:179–196, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  167. S. Pilipovic and N. Teofanov, On a symbol class of elliptic pseudodifferential operators, Bull. Cl. Sci. Math. Nat. Sci. Math., (27):57–68, 2002.

    MathSciNet  MATH  Google Scholar 

  168. S. Pilipovic and N. Teofanov, Pseudodifferential operators on ultramodulation spaces, J. Funct. Anal., 208(1):194–228, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  169. H. Rauhut, Banach frames in coorbit spaces consisting of elements which are invariant under symmetry groups, Appl. Comput. Harmon. Anal., 18(1):94–122, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  170. H. Rauhut, Coorbit space theory for quasi-Banach spaces, Preprint, 2005.

    MATH  Google Scholar 

  171. H. Rauhut, Wiener amalgam spaces with respect to quasi-Banach spaces, Preprint, 2005.

    MATH  Google Scholar 

  172. H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Clarendon Press, Oxford, 1968.

    MATH  Google Scholar 

  173. H. Reiter, Metaplectic Groups and Segal Algebras, vol. 1382 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1989.

    Book  MATH  Google Scholar 

  174. H. Reiter, On the Siegel-Weil formula, Monatsh. Math., 116(3-4):299–330, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  175. H. Reiter and J. D. Stegeman, Classical Harmonic Analysis and locally compact Groups, Second Edition, The Clarendon Press, Oxford University Press, New York, 2000.

    MATH  Google Scholar 

  176. R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases, Chap. 4 in [89], 453–488. 1998.

    MATH  Google Scholar 

  177. H.-J. Schmeisser and H. Triebel, Topics in Fourier analysis and Function Spaces, vol. 42 of Mathematik und ihre Anwendungen in Physik und Technik, Akad. Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.

    MATH  Google Scholar 

  178. M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Translation from the Russian by Stig I. Andersson, Second Edition, Springer-Verlag, Berlin, 2001.

    MATH  Google Scholar 

  179. J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett., 1(2):185–192, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  180. P. Sondergaard, Gabor frames by sampling and periodization, Adv. Comput. Math., to appear, 2006.

    Google Scholar 

  181. T. Strohmer, Numerical algorithms for discrete Gabor expansions, Chap. 8 in [89], 267–294, 1998.

    Book  MATH  Google Scholar 

  182. K. Tachizawa, The boundedness of pseudodifferential operators on modulation spaces, Math. Nachr., 168:263–277, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  183. N. Teofanov, Ultramodulation Spaces, Wilson Bases and Pseudodifferential Operators, Ph.D. Thesis, University of Novi Sad, 2000.

  184. J. Toft, Convolutions and embeddings for weighted modulation spaces, In Advances in Pseudo-Differential Operators, vol. 155 of Oper. Theory Adv. Appl., pages 165–186, Birkhöauser, Basel, 2004.

    Chapter  Google Scholar 

  185. J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I, J. Fund. Anal., 207(2):399–429, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  186. J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II, Ann. Global Anal. Geom., 26(1):73–106, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  187. J. Toft, Embeddings and compactness for generalized Sobolev-Shubin spaces and modulation spaces, Appl. Anal. 84(3):269–282, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  188. R. Tolimieri and R. S. Orr, Poisson summation, the ambiguity function, and the theory of Weyl-Heisenberg frames, J. Four. Anal. Appl., 1(3):233–247, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  189. N. Tomita, Fractional integrals on modulation spaces, Math. Nachr., 279(5-6):672–680, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  190. B. Trebels and G. Steidl, Riesz bounds of Wilson bases generated by B-splines, J. Fourier Anal. Appl., 6(2):171–184, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  191. H. Triebel, Fourier analysis and Function Spaces (Selected Topics), Teub-ner Verlagsgesellschaft, Leipzig, 1977.

    MATH  Google Scholar 

  192. H. Triebel, Spaces of Besov-Hardy-Sobolev type, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1978.

    MATH  Google Scholar 

  193. H. Triebel, Modulation spaces on the Euclidean n-space, Z. Anal. Anwendungen, 2(5):443–457, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  194. H. Triebel, The Structure of Functions, Birkhöauser, Basel, 2001.

    Book  MATH  Google Scholar 

  195. P. Wahlberg, Regularization of kernels for estimation of the Wigner spectrum of Gaussian stochastic processes with covariance in S0(R2d), Preprint, 2004.

    Google Scholar 

  196. P. Wahlberg and M. Hansson, Kernels and multiple windows for estimation of the Wigner-Ville spectrum of Gaussian locally stationary processes, Preprint, 2004.

    MATH  Google Scholar 

  197. P. Wahlberg, The random Wigner distribution of Gaussian stochastic processes with covariance in S0(R2d), J. Funct. Spaces Appl., 3(2}):163–181

    Article  MathSciNet  MATH  Google Scholar 

  198. P. Wahlberg, Vector-valued modulation spaces and localization operators with operator-valued symbols, Preprint, 2006.

    MATH  Google Scholar 

  199. D. F. Walnut, Lattice size estimates for Gabor decompositions, Monatsh. Math., 115(3):245–256, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  200. A. Weil, L'Integration dans les Groupes Topologiques et ses Applications, Hermann, 1940.

    Google Scholar 

  201. A. Weil, Sur certains groupes d'operateurs unitaires, Acta Math., 111:143–211, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  202. N. Wiener, Tauberian theorems, Ann. of Math. (2), 33(4):1–100, 1932.

    Article  MathSciNet  MATH  Google Scholar 

  203. N. Wiener, The Fourier Integral and certain of its Applications, Cambridge University Press, Cambridge, 1933.

    MATH  Google Scholar 

  204. P. Wojdyllo, Abstract Wilson systems. Part I: Theory, ESI Preprint, 2005.

    Google Scholar 

  205. M.-W. Wong, Wavelet Transforms and Localization Operators, Birk-höauser, Basel, 2002.

    Book  MATH  Google Scholar 

  206. O. Yilmaz, Coarse quantization of highly redundant time-frequency representations of square-integrable functions, Appl. Comput. Harmon. Anal., 14(2):107–132, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  207. A. C. Zaanen, Linear Analysis. North-Holland Publishing Co., Amsterdam, 1953.

    MATH  Google Scholar 

Download references

Acknowledgment

I would like to thank the guest editors of this special issue, Karlheinz Gröchenig and Christopher Heil, for their patience and endurance in soliciting this article. Finally, it was a lot of pleasure for me to summarize the very fruitful developments which modulation space theory has taken in the last twenty years. I also thank in particular Abdul Jerri for allowing me to finish this work “at the very last minute.”

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hans G. Feichtinger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Feichtinger, H.G. Modulation Spaces: Looking Back and Ahead. STSIP 5, 109–140 (2006). https://doi.org/10.1007/BF03549447

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03549447

Key words and phrases

2000 AMS Mathematics Subject Classification

Navigation