On the construction of multivariate (pre)wavelets
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
A new approach for the construction of wavelets and prewavelets onR d from multiresolution is presented. The method uses only properties of shift-invariant spaces and orthogonal projectors fromL 2(R d ) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution.
- Battle, G. (1987) A block spin construction of ondelettes, Part I: Lemarie functions. Comm. Math. Phys. 110: pp. 601-615
- Ben-Artzi, A., Ron, A. (1990) On the integer translates of a compactly supported function: dual bases and linear projectors. SIAM J. Math. Anal. 21: pp. 1550-1562
- Bennett, C., Sharpley, R. (1988) Interpolation of Operators. Academic Press, New York
- Boor, C., DeVore, R. (1983) Approximation by smooth multivariate splines. Trans. Amer. Math. Soc. 276: pp. 775-788
- [BDR]C. de Boor, R. DeVore, A. Ron (to appear):Approximation from shift-invariant subspaces of L 2(R d ). Trans. Amer. Math. Soc.
- [BDR1]C. de Boor, R. DeVore, A. Ron (to appear):The structure of finitely generated shift-invariant spaces in L 2(R d ), J. Period Functional Anal.
- Chui, C. K., Wang, J. Z. (1992) A general framework for compactly supported splines and wavelets. J. Approx. Theory 71: pp. 263-304
- Chui, C. K., Wang, J. Z. (1992) On compactly supported spline wavelets and a duality principle. Trans. Amer. Math. Soc. 330: pp. 903-912
- Chui, C. K., Stöckler, J., Ward, J. D. (1992) Compactly supported box spline wavelets. Approx. Theory & Appl. 8: pp. 77-100
- Daubechies, I. (1988) Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. XLI: pp. 909-996
- Daubechies, I. (1990) The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory 36: pp. 961-1005
- Dyn, N., Ron, A. (1990) Local approximation by certain spaces of multivariate exponential-polynomials, approximation order of exponential box splines and related interpolation problems. Trans. Amer. Math. Soc. 319: pp. 381-404
- Jia, R. Q., Micchelli, C. A. Using the refinement equation for the construction of pre-wavelets, II: powers of two. In: Laurent, P. J., Méhauté, A., Schumaker, L. L. eds. (1991) Curves and Surfaces. Academic Press, New York, pp. 209-246
- [JM1]R.-Q. Jia, C. A. Micchelli (1991): Using the Refinement Equation for the Construction of Pre-Wavelets, V: Extensibility of Trigonometric Polynomials. RC 17196, IBM.
- [JW]R.-Q. Jia, J. Z. Wang (to appear):Stability and linear independence associated with wavelet decompositions. Proc. Amer. Math. Soc.
- [LM]R. A. H. Lorentz, W. R. Madych (1991): Wavelets and generalized box splines. Arbeitspapiere der GMD No. 563.
- Mallat, S. G. (1989) Multiresolution approximations and wavelet orthonormal bases of L 2(R). Trans. Amer. Math. Soc. 315: pp. 69-87
- Meyer, Y. (1980) Ondelettes et Opérateurs I: Ondelettes. Hermann, Paris
- Micchelli, C. A. (1991) Using the refinement equation for the construction of pre-wavelets. Numer. Algorithms 1: pp. 75-116
- [Mi1]C. A. Micchelli (1991): Using the Refinement Equation for the Construction of Pre-Wavelets, IV: Cube Splines and Elliptic Splines United. Report RC 17195, IBM.
- [MRU]C. A. Micchelli, C. Rabut, F. Utreras (preprint):Using the refinement equation for the construction of pre-wavelets, III: elliptic splines.
- Riemenschneider, S. D., Shen, Z. Box splines, cardinal series, and wavelets. In: Chui, C. K. eds. (1991) Approximation Theory and Functional Analysis. Academic Press, New York, pp. 133-149
- Riemenschneider, S. D., Shen, Z. (1992) Wavelets and prewavelets in low dimensions. J. Approx. Theory 71: pp. 18-38
- Ron, A. (1988) Exponential box splines. Constr. Approx. 4: pp. 357-378
- Ron, A. (1990) Factorization theorems of univariate splines on regular grids. Israel J. Math. 70: pp. 48-68
- Rudin, W. (1974) Rela, and Complex Analysis. McGraw-Hill, New York
- Stöckler, J. Multivariate wavelets. In: Chui, C. K. eds. (1992) Wavelets—A Tutorial in Theory and Applications. Academic Press, New York, pp. 325-355
- On the construction of multivariate (pre)wavelets
Volume 9, Issue 2-3 , pp 123-166
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Primary 41A63
- Secondary 41A30
- Shift-invariant spaces
- Box splines
- Author Affiliations
- 1. Center for Mathematical Sciences, University of Wisconsin-Madison, 1308 W. Dayton Street, 53706, Madison, Wisconsin, U.S.A.
- 2. Department of Mathematics, University of South Carolina, 29208, Columbia, South Carolina, U.S.A.
- 3. Computer Sciences Department, University of Wisconsin-Madison, 1210 W. Dayton Street, 53706, Madison, Wisconsin, U.S.A.