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A characterization of semiorthogonal Parseval wavelets in abstract Hilbert spaces

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Abstract

A Parseval (multi)wavelet in L2 (ℝ) is characterized by two requirements of its Fourier transform; the characterization of a semiorthogonal Parseval wavelet requires an additional condition of the wavelet dimension function. In this article, we use the theory of generalized multiresolution analyses to extend this idea to the more general setting of an abstract Hilbert space. We find an equation that is the abstract analog of the three conditions in L2(ℝ).

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References

  1. Baggett, L., Carey, A., Moran, W., and Ohring, P. General existence theorems for orthonormal wavelets, an abstract approach,Publ. RIMS, Kyoto Univ. 31(1), 95–111, (1995).

    MATH  MathSciNet  Google Scholar 

  2. Baggett, L., Courter, J., and Merrill, K. The construction of wavelets from generalized conjugate mirror filters inL 2(ℝn),Appl. Comput. Harmon. Anal. 13(3), 201–223 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  3. Baggett, L., Jorgensen, P., Merrill, K., and Packer, J. Construction of Parseval wavelets from redundant filter systems,J. Math. Physics. 46(8), (2005).

  4. Baggett, L., Medina, H., and Merrill, K. Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ℝn,J. Fourier Anal. Appl. 5(6), 563–573, (1999).

    Article  MATH  MathSciNet  Google Scholar 

  5. Baggett, L. and Ramsay, A. A functional analytic proof of a selection lemma,Canad. J. Math. 32(2) 441–448, (1980).

    MATH  MathSciNet  Google Scholar 

  6. Bownik, M. A characterization of affine dual frames inL 2 (ℝn),Appl. Comput. Harmon. Anal. 8(2), 203–221, (2000).

    Article  MATH  MathSciNet  Google Scholar 

  7. Bownik, M. On characterizations of multiwavelets inL 2(ℝn),Proc. Amer. Math. Soc. 129(11), 3265–3274, (2001).

    Article  MATH  MathSciNet  Google Scholar 

  8. Bownik, M. Quasi-affine systems and the Calderón condition,Contemp. Math. 320, 29–43, (2003).

    MathSciNet  Google Scholar 

  9. Bownik, M. and Rzeszotnik, Z. On the existence of multiresolution analysis for framelets,Math. Ann. 332(4), 705–720, (2005).

    Article  MATH  MathSciNet  Google Scholar 

  10. Bownik, M. and Weber, E. Affine frames, GMRA’s, and the canonical dual,Studia Math. 159(3), 453–479, (2003).

    Article  MATH  MathSciNet  Google Scholar 

  11. Calogero, A. A characterization of wavelets on general lattices,J. Geom. Anal. 10(4), 579–622, (2000).

    MathSciNet  Google Scholar 

  12. Chui, C. K., Czaja, W., Maggioni, M., and Weiss, G. Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling,J. Fourier Anal. Appl. 8(2), 173–200, (2002).

    Article  MATH  MathSciNet  Google Scholar 

  13. Chui, C. K., Shi, X., and Stöckler, J. Affine frames, quasi-affine frames, and their duals,Adv. Comput. Math. 8(1–2), 1–17, (1998).

    Article  MATH  MathSciNet  Google Scholar 

  14. Cohen, A.Wavelets and Multiscale Signal Processing, Trans. R. Ryan, Chapman and Hall, London, (1995).

    MATH  Google Scholar 

  15. Courter, J. Construction of dilation-d wavelets in The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999),Contemp. Math. 247, Amer. Math. Soc. Providence, 183–206, (1999).

    Google Scholar 

  16. Federer, H. and Morse, A. P. Some properties of measurable functions,Bull. Amer. Math. Soc.,49, 270–277, (1943).

    Article  MATH  MathSciNet  Google Scholar 

  17. Folland, G.A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, (1995).

    MATH  Google Scholar 

  18. Furst, V. A characterization of semiorthogonal Parseval wavelets in abstract Hilbert spaces, Ph.D. thesis, University of Colorado at Boulder, (2006).

  19. Gripenberg, G. A necessary and sufficient condition for the existence of a father wavelet,Studia Math. 114 (3), 207–226, (1995).

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  21. Hernández, E. and Weiss, G.,A First Course on Wavelets, CRC Press, Boca Raton, (1996).

    MATH  Google Scholar 

  22. Labate, D. A unified characterization of reproducing systems generated by a finite family,J. Geom. Anal. 12(3), 469–491, (2002).

    MATH  MathSciNet  Google Scholar 

  23. Lemarié-Riensset, P. G. Analyse multi-schelles et ondelettes a support compact inLes ondelettes en 1989, Lemarié, P. G., Ed.,Lecture Notes in Math. 1438, Springer-Verlag, Berlin, 26–38, (1990).

    Google Scholar 

  24. Mallat, S. Multiresolution approximations and wavelet orthonormal bases ofL 2(ℝ),Trans. Amer. Math. Soc. 315(1), 69–87, (1989).

    Article  MATH  MathSciNet  Google Scholar 

  25. Meyer, Y.Wavelets and Operators, Cambridge Studies in Advanced Mathematics,37, Cambridge University Press, Cambridge, (1992).

    MATH  Google Scholar 

  26. Paluszyński, M., Šikić, H., Weiss, G., and Xiao, S. Generalized low pass filters and MRA frame wavelets,J. Geom. Anal. 11(2), 311–342, (2001).

    MATH  MathSciNet  Google Scholar 

  27. Paluszyński, M., Šikić, H., Weiss, G., and Xiao, S. Tight frame wavelets, their dimensions functions, MRA tight frame wavelets and connectivity properties,Adv. Comput. Math. 18(2–4), 297–327, (2003).

    Article  MATH  MathSciNet  Google Scholar 

  28. Ron, A. and Shen, Z. Affine systems inL 2(ℝd): the analysis of the analysis operatonJ. Funct. Anal. 148(2), 408–447, (1997).

    Article  MATH  MathSciNet  Google Scholar 

  29. Rzeszotnik, Z. Calderón’s condition and wavelets,Collect. Math. 52(2), 181–191, (2001).

    MATH  MathSciNet  Google Scholar 

  30. Wang, X. The study of wavelets from the properties of their Fourier transforms, Ph.D. thesis, Washington University in St. Louis, (1995).

  31. Weber, E. Frames and single wavelets for unitary groups,Canad. J. Math. 54(3), 634–647, (2002).

    MATH  MathSciNet  Google Scholar 

  32. Weiss, G. and Wilson, E. The mathematical theory of wavelets inTwentieth Century Harmonic Analysis—A Celebration (Il Ciocco, 2000), NATO Sci. Ser. II Math. Phys. Chem.33, Kluwer Acad. Publ. Dordrecht, 329–366, (2001).

    Google Scholar 

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Authors
  1. Veronika Furst
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Fort Lewis College

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Furst, V. A characterization of semiorthogonal Parseval wavelets in abstract Hilbert spaces. J Geom Anal 17, 569–591 (2007). https://doi.org/10.1007/BF02937430

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