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Wiener Estimates on Modulation Spaces

  • Joachim ToftEmail author
Chapter
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Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We characterise modulation spaces by Wiener estimates on the short-time Fourier transforms. We use the results to refine some formulae for periodic distributions with Lebesgue estimates on their coefficients.

Keywords

Wiener spaces Modulation spaces Gelfand-Shilov Quasi-Banach spaces Coorbit spaces 

Mathematics Subject Classification (2010)

Primary 42C20 43A32 42B35 46E10; Secondary 46A16 35A22 37A05 46E35 

Notes

Acknowledgements

I am very grateful to Professor Hans Feichtinger for reading parts of earlier versions of the paper and giving valuable comments, leading to improvements of the content and the style.

References

  1. 1.
    T. Aoki Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588–594.MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Chung, S.-Y. Chung, D. Kim Characterizations of the Gelfand-Shilov spaces via Fourier transforms, Proc. Amer. Math. Soc. 124 (1996), 2101–2108.MathSciNetCrossRefGoogle Scholar
  3. 3.
    H. G. Feichtinger Banach spaces of distributions of Wiener’s type and interpolation, in: Ed. P. Butzer, B. Sz. Nagy and E. Görlich (Eds), Proc. Conf. Oberwolfach, Functional Analysis and Approximation, August 1980, Int. Ser. Num. Math. 69 Birkhäuser Verlag, Basel, Boston, Stuttgart, 1981, pp. 153–165.Google Scholar
  4. 4.
    H. G. Feichtinger Modulation spaces on locally compact abelian groups. Technical report, University of Vienna, Vienna, 1983; also in: M. Krishna, R. Radha, S. Thangavelu (Eds) Wavelets and their applications, Allied Publishers Private Limited, NewDehli Mumbai Kolkata Chennai Nagpur Ahmedabad Bangalore Hyderbad Lucknow, 2003, pp. 99–140.Google Scholar
  5. 5.
    H. G. Feichtinger Modulation spaces: Looking back and ahead, Sampl. Theory Signal Image Process. 5 (2006), 109–140.MathSciNetzbMATHGoogle Scholar
  6. 6.
    H. G. Feichtinger, K. H. Gröchenig Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), 307–340.MathSciNetCrossRefGoogle Scholar
  7. 7.
    H. G. Feichtinger, K. H. Gröchenig Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsh. Math. 108 (1989), 129–148.MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. G. Feichtinger and K. H. Gröchenig Gabor frames and time-frequency analysis of distributions, J. Funct. Anal. 146 (1997), 464–495.MathSciNetCrossRefGoogle Scholar
  9. 9.
    H. G. Feichtinger, F. Luef Wiener amalgam spaces for the Fundamental Identity of Gabor Analysis, Collect. Math. 57 (2006), 233–253.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Y. V. Galperin, S. Samarah Time-frequency analysis on modulation spaces \(M^{p,q}_m\), 0 < p, q ≤, Appl. Comput. Harmon. Anal. 16 (2004), 1–18.MathSciNetCrossRefGoogle Scholar
  11. 11.
    I. M. Gelfand, G. E. Shilov Generalized functions, II-III, Academic Press, NewYork London, 1968.Google Scholar
  12. 12.
    K. H. Gröchenig Describing functions: atomic decompositions versus frames, Monatsh. Math. 112 (1991), 1–42.MathSciNetCrossRefGoogle Scholar
  13. 13.
    K. H. Gröchenig Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001.Google Scholar
  14. 14.
    K. Gröchenig Weight functions in time-frequency analysis in: L. Rodino, M. W. Wong (Eds) Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis, Fields Institute Comm. 52 2007, pp. 343–366.Google Scholar
  15. 15.
    K. H. Gröchenig, M. Leinert Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), 1–18.MathSciNetCrossRefGoogle Scholar
  16. 16.
    K. Gröchenig, G. Zimmermann Spaces of test functions via the STFT J. Funct. Spaces Appl. 2 (2004), 25–53.MathSciNetCrossRefGoogle Scholar
  17. 17.
    L. Hörmander The Analysis of Linear Partial Differential Operators, vol I–III, Springer-Verlag, Berlin Heidelberg NewYork Tokyo, 1983, 1985.Google Scholar
  18. 18.
    S. Pilipović Generalization of Zemanian spaces of generalized functions which have orthonormal series expansions, SIAM J. Math. Anal. 17 (1986), 477–484.MathSciNetCrossRefGoogle Scholar
  19. 19.
    S. Pilipović Structural theorems for periodic ultradistributions, Proc. Amer. Math. Soc. 98 (1986), 261–266.MathSciNetCrossRefGoogle Scholar
  20. 20.
    S. Pilipović Tempered ultradistributions, Boll. U.M.I. 7 (1988), 235–251.Google Scholar
  21. 21.
    H. Rauhut Wiener amalgam spaces with respect to quasi-Banach spaces, Colloq. Math. 109 (2007), 345–362.MathSciNetCrossRefGoogle Scholar
  22. 22.
    H. Rauhut Coorbit space theory for quasi-Banach spaces, Studia Math. 180 (2007), 237–253.MathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Reich A non-analytic superposition result on Gevrey-modulation spaces, Diploma thesis, Technische Universität Bergakademie Freiberg, Germany, Angewandte Mathematik Registration list 51765.Google Scholar
  24. 24.
    S. Rolewicz On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astrono. Phys. 5 (1957), 471–473.MathSciNetzbMATHGoogle Scholar
  25. 25.
    M. Reich, M. Reissig, W. Sickel Non-analytic superposition results on modulation spaces with subexponential weights, J. Pseudo-Differ. Oper. Appl. 7 (2016), 365–409.MathSciNetCrossRefGoogle Scholar
  26. 26.
    M. Ruzhansky, M. Sugimoto, J. Toft, N. Tomita Changes of variables in modulation and Wiener amalgam spaces, Math. Nachr. 284 (2011), 2078–2092.MathSciNetCrossRefGoogle Scholar
  27. 27.
    J. Toft Gabor analysis for a broad class of quasi-Banach modulation spaces in: S. Pilipović, J. Toft (eds), Pseudo-differential operators, generalized functions, Operator Theory: Advances and Applications 245, Birkhäuser, 2015, pp. 249–278.Google Scholar
  28. 28.
    J. Toft Images of function and distribution spaces under the Bargmann transform, J. Pseudo-Differ. Oper. Appl. 8 (2017), 83–139.MathSciNetCrossRefGoogle Scholar
  29. 29.
    J. Toft Semi-continuous convolutions on weakly periodic Lebesgue spaces in: P. Boggiatto, E. Cordero, M. de Gosson, H. G. Feichtinger, F. Nicola, A. Oliaro, A. Tabacco (eds), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis 95, Birkhäuser, 2019, pp. 309–321.Google Scholar
  30. 30.
    J. Toft, E. Nabizadeh Periodic distributions elements in modulation spaces, Adv. Math. 323 (2018), 193–225.MathSciNetCrossRefGoogle Scholar
  31. 31.
    B. Wang and C. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations 239 (2007), 213–250.MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsLinnæus UniversityVäxjöSweden

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