Glasnik Matematicki, Vol. 49, No. 2 (2014), 377-394.

ON TRANSLATION INVARIANT MULTIRESOLUTION ANALYSIS

Angel San Antolín

Departamento de Análisis Matemático, Universidad de Alicante, 03080 Alicante, Spain
e-mail: angel.sanantolin@ua.es

Abstract.   We give a characterization of the scaling functions and low pass filters in a translation invariant multiresolution analysis on L2(ℝ n). Our conditions involve the notion of locally non-zero function. We write our results in a general context where one considers a dilation given by a fixed expansive linear map on ℝ n preserving the integer lattice ℤ n. Indeed, for any such a linear map we construct a scaling function where the support of the Fourier transform is bounded and does not contain any open neighborhood of the origin.

2010 Mathematics Subject Classification.   42C40, 42C30, 51M15.

Key words and phrases.   Fourier transform, locally nonzero function, low pass filter, scaling function, translation invariant multiresolution analysis.

Full text (PDF) (free access)

DOI: 10.3336/gm.49.2.11

References:

  1. D. Bakić and E. N. Wilson, Scaling sets and orthonormal wavelets with dilations induced by expanding matrices, Glas. Mat. Ser. III 46(66) (2011), 189-213.
    MathSciNet     CrossRef

  2. M. Bownik, Tight frames of multidimensional wavelets, Dedicated to the memory of Richard J. Duffin. J. Fourier Anal. Appl. 3 (1997), 525-542.
    MathSciNet     CrossRef

  3. M. Bownik, Z. Rzeszotnik and D. M. Speegle, A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal. 10 (2001), 71-92.
    MathSciNet     CrossRef

  4. C. K. Chui, An introduction to wavelets, Academic Press, Inc., Boston, 1992.
    MathSciNet    

  5. P. Cifuentes, K. S. Kazarian and A. San Antolín, Characterization of scaling functions in a multiresolution analysis, Proc. Amer. Math. Soc. 133 (2005), 1013-1023.
    MathSciNet     CrossRef

  6. A. Cohen, Ondelettes, analyses multirésolutions et filtres miroirs en quadrature, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 439-459.
    MathSciNet    

  7. A. Cohen, I. Daubechies and J. C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), 485-560.
    MathSciNet     CrossRef

  8. X. Dai and D. R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc., 134 (1998), no. 640, pp. 68.
    MathSciNet     CrossRef

  9. I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, 1992.
    MathSciNet     CrossRef

  10. V. Dobrić, R. F. Gundy and P. Hitczenko, Characterizations of orthonormal scale functions: a probabilistic approach, J. Geom. Anal. 10 (2000), 417-434.
    MathSciNet     CrossRef

  11. K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of Rn, IEEE Trans. Inform. Theory 38 (1992), 556-568.
    MathSciNet     CrossRef

  12. Q. Gu and D. Han, On multiresolution analysis (MRA) wavelets in ℝ n, J. Fourier Anal. Appl. 6 (2000), 437-447.
    MathSciNet     CrossRef

  13. R. F. Gundy, Low-pass filters, martingales, and multiresolution analyses, Appl. Comput. Harmon. Anal. 9 (2000), 204-219.
    MathSciNet     CrossRef

  14. Y-H. Ha, H. Kang, J. Lee and J. K. Seo, Unimodular wavelets for L2 and the Hardy space H2, Michigan Math. J. 41 (1994), 345-361.
    MathSciNet     CrossRef

  15. E. Hernández, X. Wang and G. Weiss, Smoothing minimally supported frequency wavelets. II. J. Fourier Anal. Appl. 3 (1997), 23-41.
    MathSciNet     CrossRef

  16. E. Hernández and G. Weiss, A first course on wavelets, CRC Press, Inc. 1996.
    MathSciNet     CrossRef

  17. K. S. Kazarian and A. San Antolin, Characterization of scaling functions in a frame multiresolution analysis in HG2, in: Topics in mathematical analysis and applications, World Sci. Publ., Hackensack, 2008, 118-140.
    MathSciNet     CrossRef

  18. W. M. Lawton, Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys. 32 (1991), 57-61.
    MathSciNet     CrossRef

  19. W. M. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys. 31 (1990), 1898-1901.
    MathSciNet     CrossRef

  20. W. R. Madych, Some elementary properties of multiresolution analyses of L2(Rd), in: Wavelets, Ch. Chui ed., Academic Press, Boston, 1992, 259-294.
    MathSciNet    

  21. S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(ℝ), Trans. Amer. Math. Soc. 315 (1989), 69-87.
    MathSciNet     CrossRef

  22. Y. Meyer, Ondelettes et opérateurs. I, (French) Hermann, Paris, 1990 [English Translation: Wavelets and operators. I, Cambridge University Press, 1992.]
    MathSciNet    

  23. M. Papadakis, Unitary mappings between multiresolution analysis of L2(ℝ) and a parameterization of low-pass filters, J. Fourier Anal. Appl. 4 (1998), 199-214.
    MathSciNet     CrossRef

  24. M. Papadakis, H. Šikić and G. Weiss, The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts, J. Fourier Anal. Appl. 5 (1999), 495-521.
    MathSciNet     CrossRef

  25. A. San Antolín, Characterization of low pass filters in a multiresolution analysis, Studia Math. 190 (2009), 99-116.
    MathSciNet     CrossRef

  26. A. San Antolín, On low pass filters in a frame multiresolution analysis, Tohoku Math. J. (2) 63 (2011), 427-439.
    MathSciNet     CrossRef

  27. M. Soto-Bajo, Closure of dilates of shift-invariant subspaces, Cent. Eur. J. Math. 11 (2013), 1785-1799.
    MathSciNet     CrossRef

  28. R. S. Strichartz, Construction of orthonormal wavelets, in: Wavelets: mathematics and applications, CRC, Boca Raton, 1994, 23-50.
    MathSciNet    

  29. E. Weber, On the translation invariance of wavelet subspaces, J. Fourier Anal. Appl. 6 (2000), 551-558.
    MathSciNet     CrossRef

  30. P. Wojtaszczyk, A mathematical introduction to wavelets, Cambridge University Press, Cambridge, 1997.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page