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Multi-window dilation-and-modulation frames on the half real line

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Abstract

Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators, respectively, and have been studied extensively. However, dilation-and-modulation systems cannot be derived from wavelet or Gabor systems. This study aims to investigate a class of dilation-and-modulation systems in the causal signal space L2(ℝ+). L2(ℝ+) can be identified as a subspace of L2(ℝ), which consists of all L2(ℝ)-functions supported on ℝ+ but not closed under the Fourier transform. Therefore, the Fourier transform method does not work in L2(ℝ+). Herein, we introduce the notion of Θa-transform in L2(ℝ+) and characterize the dilation-and-modulation frames and dual frames in L2(ℝ+) using the Θa-transform; and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for L2(ℝ+). Furthermore, it has been proven that an arbitrary frame of this form is always nonredundant whenever the number of the generators is 1 and is always redundant whenever the number is greater than 1. Finally, some examples are provided to illustrate the generality of our results.

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References

  1. Albeverio S, Evdokimov S, Skopina M. p-adic multiresolution analysis and wavelet frames. J Fourier Anal Appl, 2010, 16: 693–714

    MathSciNet  MATH  Google Scholar 

  2. Casazza P G, Christensen O. Weyl-Heisenberg frames for subspaces of L2(ℝ). Proc Amer Math Soc, 2001, 129: 145–154

    MathSciNet  MATH  Google Scholar 

  3. Chen Z Y, Micchelli C A, Xu Y S. A construction of interpolating wavelets on invariant sets. Math Comp, 1999, 68: 1569–1587

    MathSciNet  MATH  Google Scholar 

  4. Christensen O. An Introduction to Frames and Riesz Bases, 2nd ed. Basel: Birkhäuser, 2016

    MATH  Google Scholar 

  5. Dahmen W, Han B, Jia R Q, et al. Biorthogonal multiwavelets on the interval: Cubic Hermite splines. Constr Approx, 2000, 16: 221–259

    MathSciNet  MATH  Google Scholar 

  6. Dai X, Diao Y, Gu Q. Subspaces with normalized tight frame wavelets in ℝ. Proc Amer Math Soc, 2001, 130: 1661–1667

    MathSciNet  MATH  Google Scholar 

  7. Dai X, Diao Y, Gu Q, et al. Frame wavelets in subspaces of L2(ℝd). Proc Amer Math Soc, 2002, 130: 3259–3267

    MathSciNet  MATH  Google Scholar 

  8. Dai X, Diao Y, Gu Q, et al. The existence of subspace wavelet sets. J Comput Appl Math, 2003, 155: 83–90

    MathSciNet  MATH  Google Scholar 

  9. Duffin R J, Schaeffer A C. A class of nonharmonic Fourier series. Trans Amer Math Soc, 1952, 72: 341–366

    MathSciNet  MATH  Google Scholar 

  10. Evdokimov S, Skopina M. On orthogonal p-adic wavelet bases. J Math Anal Appl, 2015, 424: 952–965

    MathSciNet  MATH  Google Scholar 

  11. Farkov Y A. On wavelets related to the Walsh series. J Approx Theory, 2009, 161: 259–279

    MathSciNet  MATH  Google Scholar 

  12. Farkov Y A, Maksimov A Y, Stroganov S A. On biorthogonal wavelets related to the Walsh functions. Int J Wavelets Multiresolut Inf Process, 2011, 9: 485–499

    MathSciNet  MATH  Google Scholar 

  13. Feichtinger H G, Strohmer T. Gabor Analysis and Algorithms, Theory and Applications. Boston: Birkhäuser, 1998

    MATH  Google Scholar 

  14. Feichtinger H G, Strohmer T. Advances in Gabor Analysis. Boston: Birkhäuser, 2003

    MATH  Google Scholar 

  15. Gabardo J P, Han D. Subspace Weyl-Heisenberg frames. J Fourier Anal Appl, 2001, 7: 419–433

    MathSciNet  MATH  Google Scholar 

  16. Gabardo J P, Han D. Balian-Low phenomenon for subspace Gabor frames. J Math Phys, 2004, 45: 3362–3378

    MathSciNet  MATH  Google Scholar 

  17. Gabardo J P, Han D. The uniqueness of the dual of Weyl-Heisenberg subspace frames. Appl Comput Harmon Anal, 2004, 17: 226–240

    MathSciNet  MATH  Google Scholar 

  18. Gabardo J P, Han D, Li Y Z. Lattice tiling and density conditions for subspace Gabor frames. J Funct Anal, 2013, 265: 1170–1189

    MathSciNet  MATH  Google Scholar 

  19. Gabardo J P, Li Y Z. Rational time-frequency Gabor frames associated with periodic subsets of the real line. Int J Wavelets Multiresolut Inf Process, 2014, 12: 1450013

    MathSciNet  MATH  Google Scholar 

  20. Goodman T N T, Micchelli C A, Rodriguez G, et al. On the limiting profile arising from orthonormalizing shifts of exponentially decaying functions. IMA J Numer Anal, 1998, 18: 331–354

    MathSciNet  MATH  Google Scholar 

  21. Goodman T N T, Micchelli C A, Shen Z. Riesz bases in subspaces of L2(ℝ+). Constr Approx, 2001, 17: 39–46

    MathSciNet  MATH  Google Scholar 

  22. Gröchenig K. Foundations of Time-Frequency Analysis. Boston: Birkhäuser, 2001

    MATH  Google Scholar 

  23. Gu Q, Han D. Wavelet frames for (not necessarily reducing) affine subspaces. Appl Comput Harmon Anal, 2009, 27: 47–54

    MathSciNet  MATH  Google Scholar 

  24. Gu Q, Han D. Wavelet frames for (not necessarily reducing) affine subspaces II: The structure of affine subspaces. J Funct Anal, 2011, 260: 1615–1636

    MathSciNet  MATH  Google Scholar 

  25. Hasankhani M A, Dehghan M A. A new function-valued inner product and corresponding function-valued frame in L2(0, ∞). Linear Multilinear Algebra, 2014, 8: 995–1009

    MATH  Google Scholar 

  26. Heil C. History and evolution of the density theorem for Gabor frames. J Fourier Anal Appl, 2007, 13: 113–166

    MathSciNet  MATH  Google Scholar 

  27. Heil C. A Basis Theory Primer: Expanded Edition. New York: Birkhäuser, 2011

    MATH  Google Scholar 

  28. Hernández E, Weiss G. A First Course on Wavelets. Boca Raton: CRC Press, 1996

    MATH  Google Scholar 

  29. Jia R Q, Jiang Q, Shen Z. Distributional solutions of nonhomogeneous discrete and continuous refinement equations. SIAM J Math Anal, 2000, 32: 420–434

    MathSciNet  MATH  Google Scholar 

  30. Jia R Q, Jiang Q, Shen Z. Convergence of cascade algorithms associated with nonhomogeneous refinement equations. Proc Amer Math Soc, 2001, 129: 415–427

    MathSciNet  MATH  Google Scholar 

  31. Jia H F, Li Y Z. Refinable function-based construction of weak (quasi-)affine bi-frames. J Fourier Anal Appl, 2014, 20: 1145–1170

    MathSciNet  MATH  Google Scholar 

  32. Jia H F, Li Y Z. Weak (quasi-)affine bi-frames for reducing subspaces of L2(ℝd). Sci China Math, 2015, 58: 1005–1022

    MathSciNet  MATH  Google Scholar 

  33. Lang W C. Orthogonal wavelets on the Cantor dyadic group. SIAM J Math Anal, 1996, 27: 305–312

    MathSciNet  MATH  Google Scholar 

  34. Lang W C. Wavelet analysis on the Cantor dyadic group. Houston J Math, 1998, 24: 533–544

    MathSciNet  MATH  Google Scholar 

  35. Li S. Convergence of cascade algorithms in Sobolev spaces associated with inhomogeneous refinement equations. J Approx Theory, 2000, 104: 153–163

    MathSciNet  MATH  Google Scholar 

  36. Li Y Z, Jia H F. Weak Gabor bi-frames on periodic subsets of the real line. Int J Wavelets Multiresolut Inf Process, 2015, 13: 1550046

    MathSciNet  MATH  Google Scholar 

  37. Li Y Z, Zhang W. Dilation-and-modulation systems on the half real line. J Inequal Appl, 2016, 1: 186

    MathSciNet  MATH  Google Scholar 

  38. Li Y Z, Zhang Y. Rational time-frequency vector-valued subspace Gabor frames and Balian-Low theorem. Int J Wavelets Multiresolut Inf Process, 2013, 11: 1350013

    MathSciNet  MATH  Google Scholar 

  39. Li Y Z, Zhang Y. Vector-valued Gabor frames associated with periodic subsets of the real line. Appl Math Comput, 2015, 253: 102–115

    MathSciNet  MATH  Google Scholar 

  40. Li Y Z, Zhou F Y. GMRA-based construction of framelets in reducing subspaces of L2(ℝd). Int J Wavelets Multiresolut Inf Process, 2011, 9: 237–268

    MathSciNet  MATH  Google Scholar 

  41. Micchelli C A, Xu Y. Using the matrix refinement equation for the construction of wavelets on invariant sets. Appl Comput Harmon Anal, 1994, 1: 391–401

    MathSciNet  MATH  Google Scholar 

  42. Nielsen M. Walsh-type wavelet packet expansions. Appl Comput Harmon Anal, 2000, 9: 265–285

    MathSciNet  MATH  Google Scholar 

  43. Nielsen M. Nonseparable Walsh-type functions on ℝd. Glas Mat Ser III, 2004, 39: 111–138

    MathSciNet  MATH  Google Scholar 

  44. Seip K. Regular sets of sampling and interpolation for weighted Bergman spaces. Proc Amer Math Soc, 1993, 117: 213–220

    MathSciNet  MATH  Google Scholar 

  45. Shah F A. On some properties of p-wavelet packets via the Walsh-Fourier transform. J Nonlinear Anal Optim, 2012, 3: 185–193

    MathSciNet  MATH  Google Scholar 

  46. Strang G, Zhou D X. Inhomogeneous refinement equations. J Fourier Anal Appl, 1998, 4: 733–747

    MathSciNet  MATH  Google Scholar 

  47. Sun Q. Homogeneous and nonhomogeneous refinable distributions in Fqγ. In: Wavelet Analysis and Applications, vol. 25. Providence: Amer Math Soc, 2002, 235–244

    Google Scholar 

  48. Volkmer H. Frames of wavelets in Hardy space. Analysis, 1995, 15: 405–421

    MathSciNet  MATH  Google Scholar 

  49. Young R M. An Introduction to Nonharmonic Fourier Series. New York: Academic Press, 1980

    MATH  Google Scholar 

  50. Zhang Y, Li Y Z. Rational time-frequency multi-window subspace Gabor frames and their Gabor duals. Sci China Math, 2014, 57: 145–160

    MathSciNet  MATH  Google Scholar 

  51. Zhou F Y, Li Y Z. Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of L2(ℝd). Kyoto J Math, 2010, 50: 83–99

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11271037). The authors thank the referees for their valuable comments.

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Authors and Affiliations

  1. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China

    Yunzhang Li & Wei Zhang

  2. School of Mathematics and Information Sciences, Henan University of Economics and Law, Zhengzhou, 450046, China

    Wei Zhang

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  1. Yunzhang Li
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  2. Wei Zhang
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Correspondence to Yunzhang Li.

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Li, Y., Zhang, W. Multi-window dilation-and-modulation frames on the half real line. Sci. China Math. 63, 2423–2438 (2020). https://doi.org/10.1007/s11425-018-9468-8

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  • DOI: https://doi.org/10.1007/s11425-018-9468-8

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