Skip to main content
SpringerLink
Log in

Wavelets and Other Phase Space Localization Methods

  • Conference paper
Proceedings of the International Congress of Mathematicians

Abstract

Mathematicians have various ways of judging the merits of new theorems and constructions. One very important criterion is esthetic — some developments just “feel” right, fitting, and beautiful. Just as in other venues where beauty or esthetics are discussed, taste plays an important role in this, but I think I am not alone in being especially excited when apparently different fields suddenly meet in a new concept, a new understanding. It is often of the sparks of such encounters that our esthetic enjoyment of mathematics is born.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adelson, E. H.; Simoncelli, E. P.; and Hingorani, R., 1987, Orthogonal pyramid transforms for image coding, Proc. SPIE, Cambridge, 845 (Oct.), 50–58.

    Article  Google Scholar 

  2. Andersson, L.; Hall, N.; Jawerth, B.; and Peters, G., 1994, Wavelets on closed subsets of the real line, in Schumaker, Larry L., and Webb, G. (eds.), Topics in the Theory and Applications of Wavelets. Boston: Academic Press.

    MATH  Google Scholar 

  3. Auscher, P., 1989, Ondelettes fractales et applications, Ph.D. thesis, Université de Paris IX — Dauphine, Paris, France.

    Google Scholar 

  4. Auscher, P., 1992, Toute base d’ondelettes régulières de L 2 (ℝ) est issue d’une analyse multirésolution régulière, C. R. Acad. Sci. Paris Série I, 315, 1227–1230.

    MathSciNet  MATH  Google Scholar 

  5. Battle, G., 1987, A block spin construction of ondelettes, Part I: Lemarié functions, Comm. Math. Phys., 110, 601–615.

    Article  MathSciNet  Google Scholar 

  6. Battle, G., and Federbush, P., 1993, Divergence-free wavelets, Michigan Math. J., 40, 181.

    Article  MathSciNet  Google Scholar 

  7. Beylkin, G., 1993, Wavelets and fast numerical algorithms, pp. 89–117 of Daubechies, I. (ed.), Different perspectives on wavelets. Proc. Sympos. Appl. Math., no. 47, Amer. Math. Soc., Providence, RI.

    Chapter  Google Scholar 

  8. Beylkin, G.; Coifman, R.; and Rokhlin, V., 1991, Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math., 44, 141–183.

    Article  MathSciNet  Google Scholar 

  9. Burt, P., and Adelson, E. H., 1983, The Laplacian pyramid as a compact image code, IEEE Trans. Comm., 31, 482–540.

    Article  Google Scholar 

  10. Cavaretta, A. S.; Dahmen, W.; and Micchelli, C. A., 1991, Stationary subdivision, Mem. Amer. Math. Soc., 93(453), 1–186.

    MathSciNet  MATH  Google Scholar 

  11. Chui, C. K., and Wang, J. Z., 1991, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc., 113, 785–793.

    Google Scholar 

  12. Chui, C. K., and Wang, J. Z., 1992, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc., 330, 903–915.

    MATH  Google Scholar 

  13. [13] Cohen, A., 1990, Ondelettes, analyses multirésolutions et traitement numérique du signal, Ph.D. thesis, Université Paris — Dauphine. Also Ondelettes et traitement numérique du signal, (1992), Masson, Paris; English translation to appear with Chapman and Hall, London.

    Google Scholar 

  14. Cohen, A.; Daubechies, I.; and Feauveau, J. C., 1992, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45, 485–560.

    Article  MathSciNet  Google Scholar 

  15. Cohen, A.; Daubechies, I.; and Vial, P., 1993, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmonic Anal., 1, 54–81.

    Article  MathSciNet  Google Scholar 

  16. Coifman, R., and Meyer, Y., 1991, Remarques sur l’analyse de Fourier a fenêtre, C. R. Acad. Sci. Paris Série I, 312, 259–261.

    MathSciNet  MATH  Google Scholar 

  17. Coifman, R.; Meyer, Y.; and Wickerhauser, M. V., 1992, Size properties of wavelet packets, pp. 453–470 of M. B. Ruskai, et al. (eds.), Wavelets and Their Applications, Boston: Jones and Bartlett.

    Google Scholar 

  18. Coifman, R., and Wickerhauser, M. V., 1993, Wavelets and adapted waveform analysis: a toolkit for signal processing and numerical analysis, pp. 119–153 of Daubechies, I. (ed.), Different perspectives on wavelets, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI.

    Chapter  Google Scholar 

  19. Daubechies, I., 1988, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41, 909–996.

    Article  MathSciNet  Google Scholar 

  20. Daubechies, I., and Lagarias, J. C., 1992, Two-scale difference equations, II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal., 23(4), 1031–1079.

    Article  MathSciNet  Google Scholar 

  21. Daubechies, I., and Lagarias, J. C., 1994, On the thermodynamic formalism for multifractal functions, Rev. Math. Phys., 6, 1033–1070.

    Article  MathSciNet  Google Scholar 

  22. David, G., and Journé, J. L., 1984, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2), 120, 371–397.

    Article  MathSciNet  Google Scholar 

  23. de Boor, C., 1978, A practical guide to splines, in Appl. Math. Sci. no. 27, New York: Springer.

    Google Scholar 

  24. DeVore, R. A.; Jawerth, B.; and Popov, V., 1992, Compression of wavelet decompositions, Amer. J. Math., 114, 737–785.

    Article  MathSciNet  Google Scholar 

  25. Donoho, D., 1993, Unconditional bases are optimal bases for data compression and for statistical estimation, Appl. Comput. Harmonic Anal., 1, 100–115.

    Article  MathSciNet  Google Scholar 

  26. Dyn, N.; Gregory, A.; and Levin, D., 1987, A 4-point interpolatory subdivision scheme for curve design, Comput. Aided Geom. Design, 4, 257–268.

    Article  MathSciNet  Google Scholar 

  27. Elliott, F., and Majda, A., 1994, A wavelet Monte Carlo method for turbulent diffusion with many spatial scales, J. Comp. Phys. 113, 82–109.

    Article  MathSciNet  Google Scholar 

  28. Evangelista, G., 1992, Wavelet transforms and digital filters, pp. 396–412 of Meyer, Y. (ed.), Wavelets and Applications, Paris: Masson.

    Google Scholar 

  29. Frazier, M.; Jawerth, B.; and Weiss, G., 1990, Littlewood-Paley theory and the study of function spaces, CBMS — Conference held at Auburn University in 1989, published by Amer. Math. Soc., Providence, RI.

    Google Scholar 

  30. Herley, C., and Vetterli, M., 1993, Wavelets and recursive filter banks, IEEE Trans. Signal Proc., 41, 2536–2556.

    Article  Google Scholar 

  31. [31] Jaffard, S., 1994, Multifractal formalism for functions. I & II, preprints, CERMA, Ecole Nationale des Ponts et Chaussées, Noisy-le-Grand, France.

    Google Scholar 

  32. Kelly, S. E.; Kon, M. A.; and Raphael, L. A., 1994, Local convergence for wavelet expansions, J. Funct. Anal., 126(1), 102–138.

    Article  MathSciNet  Google Scholar 

  33. Lawton, W., 1991, Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys. 32, 57–61.

    Article  MathSciNet  Google Scholar 

  34. Lemarié, P. G., 1988, Une nouvelle base d’ondelettes de L 2 (ℝ n ), J. Math. Pures Appl. (9), 67, 227–236.

    MathSciNet  Google Scholar 

  35. Lemarié, P. G., 1993, Sur l’existence des analyses multirésolutions en théorie des ondelettes, Rev. Mat. Iberoamericana, 8, 457–474.

    MATH  Google Scholar 

  36. Lemarié, P. G., and Malgouyres, G., 1989, Ondelettes sans peine, technical memorandum.

    Google Scholar 

  37. Lemarié-Rieusset, P. G., 1992, Analyses multi-resolutions non orthogonales, commutation entre projecteurs et derivation, et ondelettes vecteurs a divergence nulle, Rev. Mat. Iberoamericana, 8, 221–237.

    Article  MathSciNet  Google Scholar 

  38. Mallat, S., 1989, Multiresolution approximation and wavelet orthonormal bases of L 2 (ℝ), Trans. Amer. Math. Soc., 315, 69–88.

    MathSciNet  MATH  Google Scholar 

  39. Mallat, S., and Hwang, W. L., 1992, Singularity detection and processing with wavelets, IEEE Trans. Inform. Theory, 38, 617–643.

    Article  MathSciNet  Google Scholar 

  40. Mallat, S., and Zhang, Z., 1993, Matching pursuits with time-frequency dictionaries, IEEE Trans. on Signal Proc., 41, 3397–3415.

    Article  Google Scholar 

  41. Malvar, H., 1990, Lapped transforms for efficient transform/subband coding, IEEE Trans Acoust. Speech Signal Proc., 38, 969–978.

    Article  Google Scholar 

  42. Meyer, Y., 1985, Principe d’incertitude, bases hilbertiennes et algebres d’opérateurs, Séminaire Bourbaki, 1985–1986, no. 662.

    Google Scholar 

  43. Meyer, Y., 1990, Ondelettes et opérateurs, I: Ondelettes, II: Opérateurs de Calderón-Zygmund, III: Opérateurs multilineaires. Paris: Hermann. English translation prepared by Cambridge University Press.

    MATH  Google Scholar 

  44. Mintzer, F., 1985, Filters for distortion-free two-band multirate filter banks, IEEE Trans. Acoust. Speech Signal Proc., 33, 626–630.

    Article  Google Scholar 

  45. Rokhlin, V., 1985, Rapid solution of integral equations of classical potential theory, J. Comput. Phys., 60, 187.

    Article  MathSciNet  Google Scholar 

  46. Smith, M. J., and Barnwell, D. P., 1986, Exact reconstruction for tree-structured subband coders, IEEE Trans. Acoust. Speech Signal Proc., 34, 434–441.

    Article  Google Scholar 

  47. Stein, E., 1993, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton, NJ: Princeton University Press.

    MATH  Google Scholar 

  48. Stromberg, J. O., 1982, A modified Franklin system and higher order spline systems on ℝ n as unconditional bases for Hardy spaces, pp. 475–493 of W. Beckner, et al. (eds.), Conference in Harmonic Analysis in Honor of A. Zygmund, vol. II, Belmont, CA: Wadsworth Math. Series.

    Google Scholar 

  49. Vetterli, M., 1986, Filter banks allowing perfect reconstruction, Signal Process, 10, 219–244.

    Article  MathSciNet  Google Scholar 

  50. Wickerhauser, M. V., 1994, Adapted Wavelet Analysis from Theory to Software, Wellesley, MA: AK Peters.

    MATH  Google Scholar 

  51. Witkin, A., 1983, Scale space filtering. in Proc. Int. Joint Conf. Artificial Intell.

    Google Scholar 

  52. Wornell, G. W., and Oppenheim, A. V., 1992, Wavelet-based representations for a class of self-similar signals with application to fractal modulation, IEEE Trans. Inform. Theory, 38, 785–800.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08544, USA

    Ingrid Daubechies

Authors
  1. Ingrid Daubechies
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département de Mathématiques, EPFL, 1015, Laussane, Switzerland

    S. D. Chatterji

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Birkhäser Verlag, Basel, Switzerland

About this paper

Cite this paper

Daubechies, I. (1995). Wavelets and Other Phase Space Localization Methods. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9078-6_8

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9897-3

  • Online ISBN: 978-3-0348-9078-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation