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Transactions of the American Mathematical Society

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Multiresolution approximations and wavelet orthonormal bases of $ L\sp 2({\bf R})$

Author: Stephane G. Mallat
Journal: Trans. Amer. Math. Soc. 315 (1989), 69-87
MSC: Primary 42C10; Secondary 41A65
MathSciNet review: 1008470
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Abstract: A multiresolution approximation is a sequence of embedded vector spaces $ {({{\mathbf{V}}_j})_{j \in {\text{z}}}}$ for approximating $ {{\mathbf{L}}^2}({\mathbf{R}})$ functions. We study the properties of a multiresolution approximation and prove that it is characterized by a $ 2\pi $-periodic function which is further described. From any multiresolution approximation, we can derive a function $ \psi (x)$ called a wavelet such that $ {(\sqrt {{2^j}} \psi ({2^j}x - k))_{(k,j) \in {{\text{z}}^2}}}$ is an orthonormal basis of $ {{\mathbf{L}}^2}({\mathbf{R}})$. This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space $ {{\mathbf{H}}^s}$.

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  • [1] G. Battle, A block spin construction of ondelettes, Part 1: Lemarie functions, Comm. Math. Phys. 110 (1987), 601-615. MR 895218 (88g:81054)
  • [2] A. Cohen, Analyse multiresolutions et filtres miroirs en quadrature, Preprint, CEREMADE, Université Paris Dauphine, France.
  • [3] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. (to appear). MR 951745 (90m:42039)
  • [4] R. DeVore, The approximation of continuous functions by positive linear operators, Lecture Notes in Math., vol. 293, Springer-Verlag, 1972. MR 0420083 (54:8100)
  • [5] D. Esteban and C. Galand, Applications of quadrature mirror filters to split band voice coding schemes, Proc. Internat. Conf. Acoustic Speech and Signal Proc., May 1977.
  • [6] P. Federbush, Quantum field theory in ninety minutes, Bull. Amer. Math. Soc. 17 (1987), 93-103. MR 888881 (88k:81106)
  • [7] A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. 15 (1984), 723-736. MR 747432 (85i:81146)
  • [8] S. Jaffard and Y. Meyer, Bases d'ondelettes dans des ouverts de $ Rn$, J. Math. Pures Appl. (1987).
  • [9] R. Kronland-Martinet, J. Morlet and A. Grossmann, Analysis of sound patterns through wavelet transform, Internat. J. Pattern Recognition and Artificial Intelligence (1988).
  • [10] P. G. Lemarie, Ondelettes a localisation exponentielles, J. Math. Pures Appl. (to appear). MR 964171 (89m:42024)
  • [11] P. G. Lemarie and Y. Meyer, Ondelettes et bases Hilbertiennes, Rev. Mat. Ibero-Amer. 2 (1986). MR 864650
  • [12] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation (Tech. Rep. MS-CIS-87-22, Univ. of Pennsylvania, 1987), IEEE Trans. Pattern Analysis and Machine Intelligence, July 1989.
  • [13] Y. Meyer, Ondelletes et fonctions splines, Seminaire Equations aux Derivees Partielles, Ecole Polytechnique, Paris, France, 1986.
  • [14] -, Principe d'incertitude, bases hilbertiennes et algebres d'operateurs, Bourbaki Seminar, 1985-86, no. 662.
  • [15] M. J. Smith and T. P. Barnwell, Exact reconstruction techniques for tree-structured subband coders, IEEE Trans. Acoust. Speech Signal Process 34 (1986).
  • [16] J. Stromberg, A modified Franklin system and higher-order systems of $ {R^n}$ as unconditional bases for Hardy spaces, Conf. in Harmonic Analysis in honor of A. Zygmund, Wadsworth Math. Series, vol. 2, Wadsworth, Belmont, Calif., pp. 475-493.
  • [17] P. Tchamitchian, Biorthogonalite et theorie des operateurs, Rev. Mat. Ibero-Amer. 2 (1986). MR 990857 (91g:42014)
  • [18] -, Calcul symbolique sur les operateurs de Calderon-Zygmund et bases inconditionnelles de $ L2$ , C.R. Acad. Sci. Paris Sér. I Math. 303 (1986), 215-218. MR 860820 (88a:42023)

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Keywords: Approximation theory, orthonormal bases, wavelets
Article copyright: © Copyright 1989 American Mathematical Society

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