A geometrical solution of a problem on wavelets

Antoine Ayaghe

Studia Mathematica (2000)

  • Volume: 139, Issue: 3, page 261-273
  • ISSN: 0039-3223

Abstract

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We prove the existence of nonseparable, orthonormal, compactly supported wavelet bases for L 2 ( ℝ 2 ) of arbitrarily high regularity by using some basic techniques of algebraic and differential geometry. We even obtain a much stronger result: “most” of the orthonormal compactly supported wavelet bases for L 2 ( ℝ 2 ) , of any regularity, are nonseparable

How to cite

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Ayaghe, Antoine. "A geometrical solution of a problem on wavelets." Studia Mathematica 139.3 (2000): 261-273. <http://eudml.org/doc/216722>.

@article{Ayaghe2000,
abstract = {We prove the existence of nonseparable, orthonormal, compactly supported wavelet bases for $L^2(ℝ^2)$ of arbitrarily high regularity by using some basic techniques of algebraic and differential geometry. We even obtain a much stronger result: “most” of the orthonormal compactly supported wavelet bases for $L^2(ℝ^2)$, of any regularity, are nonseparable},
author = {Ayaghe, Antoine},
journal = {Studia Mathematica},
keywords = {bivariate wavelets; non-separable wavelet basis; orthonormal wavelet basis; compact support; wavelets of high regularity},
language = {eng},
number = {3},
pages = {261-273},
title = {A geometrical solution of a problem on wavelets},
url = {http://eudml.org/doc/216722},
volume = {139},
year = {2000},
}

TY - JOUR
AU - Ayaghe, Antoine
TI - A geometrical solution of a problem on wavelets
JO - Studia Mathematica
PY - 2000
VL - 139
IS - 3
SP - 261
EP - 273
AB - We prove the existence of nonseparable, orthonormal, compactly supported wavelet bases for $L^2(ℝ^2)$ of arbitrarily high regularity by using some basic techniques of algebraic and differential geometry. We even obtain a much stronger result: “most” of the orthonormal compactly supported wavelet bases for $L^2(ℝ^2)$, of any regularity, are nonseparable
LA - eng
KW - bivariate wavelets; non-separable wavelet basis; orthonormal wavelet basis; compact support; wavelets of high regularity
UR - http://eudml.org/doc/216722
ER -

References

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  2. [A1] A. Ayache, Construction of non-separable dyadic compactly supported orthonormal wavelet bases for L 2 ( ℝ 2 ) of arbitrarily high regularity, Rev. Mat. Iberoamericana 15 (1999), 37-58. Zbl0923.42021
  3. [A2] A. Ayache, Bases multivariées d'ondelettes, orthonormales, non séparables, à support compact et de régularité arbitraire, Phd Thesis, Ceremade, Univ. Paris Dauphine, 1997. 
  4. [BW] E. Belogay and Y. Wang, Arbitrarily smooth orthogonal nonseparable wavelets in ℝ 2 , SIAM J. Math. Anal. 30 (1999), 678-697. Zbl0946.42025
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  8. [D] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996. Zbl0644.42026
  9. [J] R. Q. Jia, Characterization of smoothness of multivariate refinable functions in sobolev spaces, preprint. Zbl1052.42029
  10. [KLe] J. P. Kahane and P. G. Lemarié-Rieusset, Fourier Series and Wavelets, Gordon and Breach, 1995. 
  11. [L] W. M. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys. 31 (1990), 1898-1901. Zbl0708.46020
  12. [LR] W. M. Lawton and H. L. Resnikoff, Multidimensional wavelet bases, preprint, 1991. 
  13. [M] Y. Meyer, Ondelettes et opérateurs, Hermann, 1990. Zbl0694.41037
  14. [RW] H. L. Resnikoff and R. O. Wells Jr., Wavelet Analysis: The Scalable Structure of Information, Springer, 1998. 
  15. [W] R. O. Wells Jr., Parametrizing smooth compactly supported wavelets, Trans. Amer. Math. Soc. 338 (1993), 919-931. Zbl0777.41029

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