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Explicit constructions and properties of generalized shift-invariant systems in \(L^{2}(\mathbb {R})\)

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Abstract

Generalized shift-invariant (GSI) systems, originally introduced by Hernández et al. and Ron and Shen, provide a common frame work for analysis of Gabor systems, wavelet systems, wave packet systems, and other types of structured function systems. In this paper we analyze three important aspects of such systems. First, in contrast to the known cases of Gabor frames and wavelet frames, we show that for a GSI system forming a frame, the Calderón sum is not necessarily bounded by the lower frame bound. We identify a technical condition implying that the Calderón sum is bounded by the lower frame bound and show that under a weak assumption the condition is equivalent with the local integrability condition introduced by Hernández et al. Second, we provide explicit and general constructions of frames and dual pairs of frames having the GSI-structure. In particular, the setup applies to wave packet systems and in contrast to the constructions in the literature, these constructions are not based on characteristic functions in the Fourier domain. Third, our results provide insight into the local integrability condition (LIC).

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References

  1. Balazs, P., Dörfler, M., Jaillet, F., Holighaus, N., Velasco, G.: Theory, implementation and applications of nonstationary Gabor frames. J. Comput. Appl. Math. 236(6), 1481–1496 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bownik, M., Lemvig, J.: Affine and quasi-affine frames for rational dilations. Trans. Amer. Math. Soc. 363(4), 1887–1924 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bownik, M., Rzeszotnik, Z.: The spectral function of shift-invariant spaces on general lattices. In: Wavelets, Frames and Operator Theory, Volume 345 of Contemp. Math., pp 49–59. Amer. Math. Soc., Providence (2004)

  4. Christensen, O.: An Introduction to Frames and Riesz Bases. Second Expanded Edition. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston (2016)

    Google Scholar 

  5. Christensen, O., Goh, S. S.: Fourier-like frames on locally compact abelian groups. J. Approx. Theory 192, 82–101 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Christensen, O., Kim, R. Y.: On dual Gabor frame pairs generated by polynomials. J. Fourier Anal. Appl. 16(1), 1–16 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Christensen, O., Osgooei, E.: On frame properties for Fourier-like systems. J. Approx. Theory 172, 47–57 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Christensen, O., Rahimi, A.: Frame properties of wave packet systems in L 2(ℝ d). Adv. Comput. Math. 29(2), 101–111 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Christensen, O., Sun, W.: Explicitly given pairs of dual frames with compactly supported generators and applications to irregular B-splines. J. Approx. Theory 151 (2), 155–163 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chui, C. K., Shi, X. L.: Inequalities of Littlewood-Paley type for frames and wavelets. SIAM J. Math. Anal. 24(1), 263–277 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Córdoba, A., Fefferman, C.: Wave packets and fourier integral operators. Comm. Partial Differ. Equ. 3(11), 979–1005 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  12. Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics. Philadelphia (1992)

  13. Gröchenig, K.: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston (2001)

    MATH  Google Scholar 

  14. Hernández, E., Labate, D., Weiss, G.: A unified characterization of reproducing systems generated by a finite family. II. J. Geom. Anal. 12(4), 615–662 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hernández, E., Labate, D., Weiss, G., Wilson, E.: Oversampling, quasi-affine frames, and wave packets. Appl. Comput. Harmon. Anal. 16(2), 111–147 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jakobsen, M. S., Lemvig, J.: Reproducing formulas for generalized translation invariant systems on locally compact abelian groups. Trans. Amer. Math. Soc. doi:10.1090/tran/6594, to appear in print (2016)

  17. Kutyniok, G., Labate, D.: The theory of reproducing systems on locally compact abelian groups. Colloq. Math. 106(2), 197–220 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Labate, D., Weiss, G., Wilson, E.: An approach to the study of wave packet systems. In: Wavelets, Frames and Operator Theory, Volume 345 of Contemp. Math., pp 215–235. Amer. Math. Soc., Providence (2004)

  19. Lemvig, J.: Constructing pairs of dual bandlimited framelets with desired time localization. Adv. Comput. Math. 30(3), 231–247 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lemvig, J.: Constructing pairs of dual bandlimited frame wavelets in L 2(ℝ d). Appl. Comput. Harmon. Anal. 32(3), 313–328 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ron, A., Shen, Z.: Generalized shift-invariant systems. Constr. Approx. 22 (1), 1–45 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yang, X., Zhou, X.: An extension of the Chui-Shi frame condition to nonuniform affine operations. Appl. Comput. Harmon. Anal. 16(2), 148–157 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, 2800, Kgs. Lyngby, Denmark

    Ole Christensen, Marzieh Hasannasab & Jakob Lemvig

Authors
  1. Ole Christensen
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  2. Marzieh Hasannasab
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  3. Jakob Lemvig
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Correspondence to Ole Christensen.

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Communicated by: Yang Wang

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Christensen, O., Hasannasab, M. & Lemvig, J. Explicit constructions and properties of generalized shift-invariant systems in \(L^{2}(\mathbb {R})\) . Adv Comput Math 43, 443–472 (2017). https://doi.org/10.1007/s10444-016-9492-x

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  • DOI: https://doi.org/10.1007/s10444-016-9492-x

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