Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Stephane G. Mallat
Journal: Trans. Amer. Math. Soc. 315 (1989), 69-87
MSC: Primary 42C10; Secondary 41A65
DOI: https://doi.org/10.1090/S0002-9947-1989-1008470-5
MathSciNet review: 1008470
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 ${{\mathbf {H}}^s}$.


References [Enhancements On Off] (What's this?)

  • Guy Battle, A block spin construction of ondelettes. I. Lemarié functions, Comm. Math. Phys. 110 (1987), no. 4, 601–615. MR 895218
  • A. Cohen, Analyse multiresolutions et filtres miroirs en quadrature, Preprint, CEREMADE, Université Paris Dauphine, France.
  • Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909–996. MR 951745, DOI https://doi.org/10.1002/cpa.3160410705
  • Ronald A. DeVore, The approximation of continuous functions by positive linear operators, Lecture Notes in Mathematics, Vol. 293, Springer-Verlag, Berlin-New York, 1972. MR 0420083
  • 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.
  • Paul Federbush, Quantum field theory in ninety minutes, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 93–103. MR 888881, DOI https://doi.org/10.1090/S0273-0979-1987-15521-2
  • A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), no. 4, 723–736. MR 747432, DOI https://doi.org/10.1137/0515056
  • S. Jaffard and Y. Meyer, Bases d’ondelettes dans des ouverts de $Rn$, J. Math. Pures Appl. (1987). R. Kronland-Martinet, J. Morlet and A. Grossmann, Analysis of sound patterns through wavelet transform, Internat. J. Pattern Recognition and Artificial Intelligence (1988).
  • Pierre Gilles Lemarié, Ondelettes à localisation exponentielle, J. Math. Pures Appl. (9) 67 (1988), no. 3, 227–236 (French, with English summary). MR 964171
  • P. G. Lemarié and Y. Meyer, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2 (1986), no. 1-2, 1–18 (French). MR 864650, DOI https://doi.org/10.4171/RMI/22
  • 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. Y. Meyer, Ondelletes et fonctions splines, Seminaire Equations aux Derivees Partielles, Ecole Polytechnique, Paris, France, 1986. ---, Principe d’incertitude, bases hilbertiennes et algebres d’operateurs, Bourbaki Seminar, 1985-86, no. 662. M. J. Smith and T. P. Barnwell, Exact reconstruction techniques for tree-structured subband coders, IEEE Trans. Acoust. Speech Signal Process 34 (1986). 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.
  • Philippe Tchamitchian, Biorthogonalité et théorie des opérateurs, Rev. Mat. Iberoamericana 3 (1987), no. 2, 163–189 (French). MR 990857, DOI https://doi.org/10.4171/RMI/48
  • Philippe Tchamitchian, Calcul symbolique sur les opérateurs de Calderón-Zygmund et bases inconditionnelles de $L^2({\bf R})$, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 6, 215–218 (French, with English summary). MR 860820

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42C10, 41A65

Retrieve articles in all journals with MSC: 42C10, 41A65


Additional Information

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