Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-20T04:30:11.510Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiresolution and wavelets

Published online by Cambridge University Press:  20 January 2009

Rong-Qing Jia  and
Zuowei Shen
Rong-Qing Jia
Affiliation:
Department of MathematicsUniversity of AlbertaEdmonton, CanadaT6G 2G1
Zuowei Shen
Affiliation:
Center for Mathematical SciencesUniversity of WisconsinMadison, WI 53705, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Multiresolution is investigated on the basis of shift-invariant spaces. Given a finitely generated shift-invariant subspace S of L2(ℝd), let Sk be the 2k-dilate of S (k∈ℤ). A necessary and sufficient condition is given for the sequence {Sk}k∈ℤ to fom a multiresolution of L2(ℝd). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skew-symmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 37 , Issue 2 , June 1994 , pp. 271 - 300
Copyright
Copyright © Edinburgh Mathematical Society 1994

References

REFERENCES

1.Adams, J. F., Vector fields on spheres, Ann. of Math. 75 (1962), 603–632.CrossRefGoogle Scholar
2.Battle, G., A block spin construction of ondelettes. Part I: Lemaire Functions, Comm. Math. Phys. 110 (1987), 601–615.CrossRefGoogle Scholar
3.de Boor, C. and Höllig, K., B-splines from parallelepipeds, J. Analyse Math. 42 (1982/1983), 99–115.CrossRefGoogle Scholar
4.de Boor, C., DeVore, R. and Ron, A., Approximation from shift-invariant subspaces of L 2(ℝd), (CMS-TSR University of Wisconsin-Madison 92–2, 1991).CrossRefGoogle Scholar
5.de Boor, C., DeVore, R. and Ron, A., The structure of finitely generated shift-invariant spaces in L 2(ℝd), (CMS-TSR University of Wisconsin-Masison 92–8, 1992).CrossRefGoogle Scholar
6.de Boor, C., DeVore, R. and Ron, A., On the construction of multivariate (pre) wavelets, Constr. Approx. 9 (1993), 123–166.CrossRefGoogle Scholar
7.Chui, C. K. and Wang, J. Z., On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992), 903–915.CrossRefGoogle Scholar
8.Chui, C. K. and Wang, J. Z., A general framework of compactly supported splines and wavelets, J. Approx. Theory 71 (1992), 263–304.CrossRefGoogle Scholar
9.Chui, C. K., Stöckler, J. and Ward, J. D., Compactly supported box spline wavelets, Approx. Theory Appl. 8 (1992), 77–100.CrossRefGoogle Scholar
10.Ciarlet, P. G., Introduction to Numerical Linear Algebra and Optimisation (Cambridge, University Press, 1988).Google Scholar
11.Ciesielski, Z., Equivalence, unconditionality and convergence a.e. of the spline bases in L p spaces, in Approximation Theory (Banach Center Publications, 4) 55–68.Google Scholar
12.Cohen, A., Ondelettes, analysis multiresolutions et traitement numerique du signal (Ph.D. Thesis, Universite de Paris IX (Dauphine), France, 1990).Google Scholar
13.Dahmen, W. and Kunoth, A., Multilevel preconditioning (Preprint No. A-91–31, Fachbereich Mathematik, Freie Universität Berlin, 1991).Google Scholar
14.Dahmen, W. amd Micchelli, C. A., Translates of multivariate splines, Linear Algebra Appl. 52/53 (1983), 217–234.Google Scholar
15.Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909–996.CrossRefGoogle Scholar
16.Daubechies, I., Ten Lectures on Wavelets (CBMF Conferences Series in Applied Mathematics, 61, SIAM, Philadelphia, 1992).CrossRefGoogle Scholar
17.DeVore, R. and Lucier, B., Wavelets, Acta Numerica 1 (1991), 1–56.CrossRefGoogle Scholar
18.Goodman, T. N. T., Lee, S. L. and Wang, W. S., Wavelets in wandering subspaces, Trans. Amer. Math. Soc. 338 (1993), 639–654.CrossRefGoogle Scholar
19.Gröchenig, K., Analyse multi-échelles et bases d'ondelettes, C. R. Acad. Sci. Paris Sér. I Math. 305(1) (1987), 13–15.Google Scholar
20.Helson, H., Lectures on Invariant Subspaces (Academic Press, New York, 1964).Google Scholar
21.James, I. M., The Topology of Stiefel Manifolds (LMS Lecture Note Series 24, Cambridge University Press, Cambridge, 1976).Google Scholar
22.Jia, R. Q., On the linear independence of translates of a box spline, J. Approx. Theory 40 (1984), 158–160.CrossRefGoogle Scholar
23.Jia, R. Q., A Bernstein type inequality associated with wavelet decomposition, Constr. Approx. 9 (1993), 299–318.CrossRefGoogle Scholar
24.Jia, R. Q. and Micchelli, C. A., On linear independence of integer translates of a finite number of functions (Research Report CS-90–10, 1990, University of Waterloo). A revised version appeared i. Proc. Edinburgh Math. Soc 36 (1992), 69–85.CrossRefGoogle Scholar
25.Jia, R. Q. and Micchelli, C. A., Using the refinement equation for the construction of pre-wavelets II: Powers of two, in Curves and Surfaces (Laurent, P. J., Le Méhauté, A. and Schumaker, L. L., eds.) Academic Press, New York, 1991), 209–246.CrossRefGoogle Scholar
26.Jia, R. Q. and Micchelli, C. A., Using the refinement equation for the construction of pre-wavelets V: extensibility of trigonometric polynomials, Computing 48 (1992), 61–72.CrossRefGoogle Scholar
27.Jia, R. Q. and Wang, J. Z., Stability and linear independence associated with wavelet decompositions, Proc. Amer. Math. Soc. 117 (1993), 1115–1124.CrossRefGoogle Scholar
28.Lam, T. Y., Serre's Conjecture (Lecture Notes in Math. 635, Springer-Verlag, New York, 1978).CrossRefGoogle Scholar
29.Lemarié, P. G., Ondelettes à localisation exponentielle, J. Math. Pures Appl. 67 (1988), 227–236.Google Scholar
30.Lorentz, R. A. and Madych, W. R., Wavelets and generalized box splines, preprint.Google Scholar
31.Madych, W. R., Some elementary properties of multiresolution analyses of L 2(ℝn), in Wavelets—A Tutorial in Theory and Applications (Academic Press, New York, 1992), 259–294.Google Scholar
32.Mallat, S. G., Multiresolution approximations and wavelet orthonormal bases of L 2(ℝn), Trans. Amer. Math. Soc. 315 (1989), 69–87.Google Scholar
33.Meyer, Y., Ondelettes et Opérateurs I: in Ondelettes (Herman Éditeurs, 1990).Google Scholar
34.Micchelli, C. A., Using the refinement equation for the construction of pre-wavelets, Numerical Algorithms 1 (1991), 75–116.CrossRefGoogle Scholar
35.Micchelli, C. A., Using the refinement equation for the construction of pre-wavelets VI: shift-invariant subspaces, to appear in the Proceedings of the NATO ASI held in Maratea, Italy, 04 1991.CrossRefGoogle Scholar
36.Micchelli, C. A., A tutorial on multivariate wavelet decomposition, to appear in the Proceedings of the NATO ASI held in Maratea, Italy, 04 1991.Google Scholar
37.Riemenschneider, S. D. and Shen, Z. W., Box splines, cardinal series and wavelets, in Approximation Theory and Functional Analysis (Chui, C. K., ed., Academic Press, 1991), 133–149.Google Scholar
38.Riemenschneider, S. D. and Shen, Z. W., Wavelets and prewavelets in low dimensions, J. Approx. Theory 71 (1992), 18–38.CrossRefGoogle Scholar
39.Stöckler, J., Multivariate wavelets, in Wavelets—A Tutorial in Theory and Applications (Academic Press, New York, 1992), 325–355.Google Scholar
40.Strömberg, J.-O., A modified Franklin system and higher-order spline system on ℝn as unconditional bases for Hardy spaces, in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. II, Beckner, W. et al. , eds., Wadsworth, Belmont, California, 1983), 457–493.Google Scholar
41.Wiener, N., The Fourier Integral and Certain of its Applications (Dover Publications, Inc., New York, 1958).Google Scholar