Stockwell-like frames for Sobolev spaces

  • Ubertino BattistiEmail author
  • Michele Berra
  • Anita Tabacco


We construct a family of frames describing the norm and seminorm of the space \(H^s(\mathbb {R}^d)\). We also characterise Besov spaces modeled on \(L^2(\mathbb {R}^d)\). Our work is inspired by the discrete orthonormal Stockwell transform introduced by R.G. Stockwell, which provides a time-frequency localised version of the Fourier basis of \(L^2([0,1])\). This approach is a hybrid between Gabor and Wavelet frames. We construct explicit and computable examples of these frames, discussing their properties and comparing them with the existing literature.


Frames Stockwell transform Sobolev spaces Decomposition spaces 

Mathematics Subject Classification

42C15 42C40 46E35 



We thank Fabio Nicola and Sandra Saliani for useful discussions on the subject. We also acknowledge the anonymous referee who helped improving the quality of the paper. We acknowledge that the present research has been partially supported by MIUR grant Dipartimenti di Eccellenza 2018-2022.


  1. 1.
    Battisti, U., Riba, L.: Window-dependent bases for efficient representations of the Stockwell transform. Appl. Comput. Harmonic Anal. 40(2), 292–320 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Battisti, U., Berra, M.: Explicit examples of \(L^2\)-frames associated to \(\alpha \)-partitioning. arXiv:1509.01437
  3. 3.
    Bagher-Ebadian, H., Siddiqui, F., Liu, C., Movsas, B., Chetty, I.J.: On the impact of smoothing and noise on robustness of CT and CBCT radiomics features for patients with head and neck cancers. Med. Phys. 44(5), 1755–1770 (2017)CrossRefGoogle Scholar
  4. 4.
    Berra, M., de Hoop, M.V., Romero, J.L.: A multi-scale Gaussian beam parametrix for the wave equation: the Dirichlet boundary value problem. arXiv:1705.00337
  5. 5.
    Borup, L., Nielsen, M.: Frame decomposition of decomposition spaces. J. Fourier Anal. Appl. 13(1), 39–70 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cabrelli, C., Mosquera, C.A., Paternostro, V.: Linear combinations of frame generators in systems of translates. J. Math. Anal. Appl. 413(2), 776–788 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Candès, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise \(C^2\) singularities. Commun. Pure Appl. Math. 57(2), 219–266 (2004)CrossRefGoogle Scholar
  8. 8.
    Canuto, C., Tabacco, A.: Multilevel decompositions of functional spaces. J. Fourier Anal. Appl. 3(6), 715–742 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston (2016)zbMATHGoogle Scholar
  10. 10.
    Daubechies, I.: Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992)Google Scholar
  11. 11.
    Drabycz, S., Stockwell, R.G., Mitchell, J.R.: Image texture characterization using the discrete orthonormal S-transform. J. Digit. Imaging 22(6), 696–708 (2008)CrossRefGoogle Scholar
  12. 12.
    Feichtinger, H.G.: Atomic characterizations of modulation spaces through Gabor-type representations. Rocky Mt. J. Math. 19(1), 113–125 (1989). Constructive Function Theory—86 Conference (Edmonton, AB, 1986)Google Scholar
  13. 13.
    Feichtinger, H.G., Gröbner, P.: Banach spaces of distributions defined by decomposition methods. I. Math. Nachr. 123, 97–120 (1985)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fornasier, M.: Banach frames for \(\alpha \)-modulation spaces. Appl. Comput. Harmonic Anal. 22(2), 157–175 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grobner, P.: Banachraeume Glatter Funktionen und Zerlegungsmethoden. ProQuest LLC, Ann Arbor, MI (1992). Thesis (Dr.natw.)—Technische Universitaet Wien (Austria)Google Scholar
  16. 16.
    Gröchenig, K.: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston (2001)zbMATHGoogle Scholar
  17. 17.
    Guo, Q., Molahajloo, S., Wong, M.W.: Phases of modified Stockwell transforms and instantaneous frequencies. J. Math. Phys. 51(5), 05210-11 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jansen, M.: Non-equispaced B-spline wavelets. Int. J. Wavelets Multiresolut. Inf. Process. 14(6), 1650056 (2016). (35) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lions, J.-L., Peetre, J.: Sur une classe d’espaces d’interpolation. Inst. Hautes Études Sci. Publ. Math. 19, 5–68 (1964)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, New York (1999)zbMATHGoogle Scholar
  21. 21.
    Ottosen, E.S., Nielsen, M.: A characterization of sparse nonstationary Gabor expansions. J. Fourier Anal. Appl. (2017). CrossRefGoogle Scholar
  22. 22.
    Pridham, G., Steenwijk, M.D., Geurts, J.J.G., Zhang, Y.: A discrete polar Stockwell transform for enhanced characterization of tissue structure using MRI. Magn. Reson. Med. 44, 1755–1770 (2017)Google Scholar
  23. 23.
    Riba, L., Wong, M.W.: Continuous inversion formulas for multi-dimensional Stockwell transforms. Math. Model. Nat. Phenom. 8(1), 215–229 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Starck, J.-L., Candès, E.J., Donoho, D.L.: The curvelet transform for image denoising. IEEE Trans. Image Process. 11(6), 670–684 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Stockwell, R.G.: A basis for efficient representation of the S-transform. Digit. Signal Process. 17, 371–393 (2007)CrossRefGoogle Scholar
  26. 26.
    Stockwell, R.G., Mansinha, L., Lowe, R.P.: Localization of the complex spectrum: the S transform. IEEE Trans. Signal Process. 44, 998–1001 (1996)CrossRefGoogle Scholar
  27. 27.
    Voigtlaender, F.: Structured, compactly supported Banach frame decompositions of decomposition spaces (2016). arXiv:1612.08772
  28. 28.
    Wang, Y., Orchard, J.: Fast discrete orthonormal Stockwell transform. SIAM J. Sci. Comput. 31(5), 4000–4012 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wong, M.W., Zhu, H.: A characterization of Stockwell spectra. In: Modern Trends in Pseudo-Differential Operators, vol. 172 of Operator Theory: Advances and Applications, pp. 251–257. Birkhäuser, Basel (2007)Google Scholar
  30. 30.
    Yan, Y., Zhu, H.: The generalization of discrete Stockwell transforms. In: 9th European Signal Processing Conference, pp. 1209–1213 (2011)Google Scholar
  31. 31.
    Zhu, H., Goodyear, B.G., Lauzon, M.L., Brown, R.A., et al.: A new local multiscale fourier analysis for medical imaging. Med. Phys. 30(6), 1134–1141 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Politecnico di TorinoTurinItaly
  2. 2.MIURCavallermaggioreItaly

Personalised recommendations