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Weak (quasi-)affine bi-frames for reducing subspaces of L 2(ℝd)

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Abstract

Since a frame for a Hilbert space must be a Bessel sequence, many results on (quasi-)affine bi-frame are established under the premise that the corresponding (quasi-)affine systems are Bessel sequences. However, it is very technical to construct a (quasi-)affine Bessel sequence. Motivated by this observation, in this paper we introduce the notion of weak (quasi-)affine bi-frame (W(Q)ABF) in a general reducing subspace for which the Bessel sequence hypothesis is not needed. We obtain a characterization of WABF, and prove the equivalence between WABF and WQABF under a mild condition. This characterization is used to recover some related known results in the literature.

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References

  1. Benedetto J J, Li S. The theory of multiresolution analysis frames and applications to filter banks. Appl Comput Harmon Anal, 1998, 5:389–427

    Article  MATH  MathSciNet  Google Scholar 

  2. Bownik M. Tight frames of multidimensional wavelets. J Fourier Anal Appl, 1997, 3:525–542

    Article  MATH  MathSciNet  Google Scholar 

  3. Bownik M. A characterization of affine dual frames in L 2(ℝn). Appl Comput Harmon Anal, 2000, 8:203–221

    Article  MATH  MathSciNet  Google Scholar 

  4. Bownik M. Anisotropic Hardy spaces and wavelets. Mem Amer Math Soc, 2003, 164:122pp

  5. Christensen O. An Introduction to Frames and Riesz Bases. Boston: Birkhäuser, 2003

    Book  MATH  Google Scholar 

  6. Chui C K, He W. Compactly supported tight frames associated with refinable functions. Appl Comput Harmon Anal, 2000, 8:293–319

    Article  MATH  MathSciNet  Google Scholar 

  7. Chui CK, He W, Stöckler J. Compactly supported tight and sibling frames with maximum vanishing moments. Appl Comput Harmon Anal, 2002, 13:224–262

    Article  MATH  MathSciNet  Google Scholar 

  8. Chui C K, Shi X, Stöckler J. Affine frames, quasi-affine frames, and their duals. Adv Comput Math, 1998, 8:1–17

    Article  MATH  MathSciNet  Google Scholar 

  9. Chui C K, Shi X. Bessel sequences and affine frames. Appl Comput Harmon Anal, 1993, 1:29–49

    Article  MATH  MathSciNet  Google Scholar 

  10. Dai X, Diao Y, Gu Q, et al. Frame wavelets in subspaces of L 2(ℝd). Proc Amer Math Soc, 2002, 130:3259–3267

    Article  MATH  MathSciNet  Google Scholar 

  11. Daubechies I, Han B, Ron A, et al. Framelets, MRA-based constructions of wavelet frames. Appl Comput Harmon Anal, 2003, 14:1–46

    Article  MATH  MathSciNet  Google Scholar 

  12. Ehler M. On multivariate compactly supported bi-frames. J Fourier Anal Appl, 2007, 13:511–532

    Article  MATH  MathSciNet  Google Scholar 

  13. Ehler M. Compactly supported multivariate pairs of dual wavelet frames obtained by convolution. Int J Wavelets Multiresolut Inf Process, 2008, 6:183–208

    Article  MATH  MathSciNet  Google Scholar 

  14. Ehler M. Nonlinear approximation associated with nonseparable wavelet bi-frames. J Approx Theory, 2009, 161:292–313

    Article  MATH  MathSciNet  Google Scholar 

  15. Ehler M. The multiresolution structure of pairs of dual wavelet frames for a pair of Sobolev spaces. Jaen J Approx, 2010, 2:193–214

    MATH  MathSciNet  Google Scholar 

  16. Ehler M, Han B. Wavelet bi-frames with few generators from multivariate refinable functions. Appl Comput Harmon Anal, 2008, 25:407–414

    Article  MATH  MathSciNet  Google Scholar 

  17. Frazier M, Garrigdós G, Wang K, et al. A characterization of functions that generate wavelet and related expansion. J Fourier Anal Appl, 1997, 3:883–906

    Article  MATH  MathSciNet  Google Scholar 

  18. Gröchenig K, Ron A. Tight compactly supported wavelet frames of arbitrarily high smoothness. Proc Amer Math Soc, 1998, 126:1101–1107

    Article  MATH  MathSciNet  Google Scholar 

  19. Han B, Shen Z. Dual wavelet frames and Riesz bases in Sobolev spaces. Constr Approx, 2009, 29:369–406

    Article  MATH  MathSciNet  Google Scholar 

  20. Jiang Q. Wavelet bi-frames with uniform symmetry for curve multiresolution processing. J Comput Appl Math, 2011, 235:1653–1675

    Article  MATH  MathSciNet  Google Scholar 

  21. Li Y Z, Zhou F Y. Affine and quasi-affine dual wavelet frames in reducing subspaces of L 2(ℝd). Acta Math Sinica Chin Ser, 2010, 53:551–562

    MATH  MathSciNet  Google Scholar 

  22. Li Y Z, Zhou F Y. GMRA-based construction of framelets in reducing subspaces of L 2(ℝd). Int JWavelets Multiresolut Inf Process, 2011, 9:237–268

    Article  MATH  Google Scholar 

  23. Lian Q F, Li Y Z. Reducing subspace frame multiresolution analysis and frame wavelets. Commun Pure Appl Anal, 2007, 6:741–756

    Article  MATH  MathSciNet  Google Scholar 

  24. Petukhov A P. Explicit construction of framelets. Appl Comput Harmon Anal, 2001, 11:313–327

    Article  MATH  MathSciNet  Google Scholar 

  25. Ron A, Shen Z. Affine systems in L 2(ℝd): the analysis of the analysis operator. J Funct Anal, 1997, 148:408–447

    Article  MATH  MathSciNet  Google Scholar 

  26. Ron A, Shen Z. Affine systems in L 2(ℝd) II: Dual systems. J Fourier Anal Appl, 1997, 3:617–637

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    HuiFang Jia & YunZhang Li

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  1. HuiFang Jia
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  2. YunZhang Li
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Correspondence to YunZhang Li.

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Jia, H., Li, Y. Weak (quasi-)affine bi-frames for reducing subspaces of L 2(ℝd). Sci. China Math. 58, 1005–1022 (2015). https://doi.org/10.1007/s11425-014-4906-z

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  • DOI: https://doi.org/10.1007/s11425-014-4906-z

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