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.
- [B]G. Battle (1987):A block spin construction of ondelettes, Part I: Lemarie functions. Comm. Math. Phys.,110:601–615.
- [BR]A. Ben-Artzi, A. Ron (1990):On the integer translates of a compactly supported function: dual bases and linear projectors SIAM J. Math. Anal.,21:1550–1562.
- [BS]C. Bennett, R. Sharpley (1988): Interpolation of Operators. Pure and Applied Mathematics, vol. 129. New York: Academic Press.
- [BD]C. de Boor, R. DeVore (1983):Approximation by smooth multivariate splines. Trans. Amer. Math. Soc.,276:775–788.
- [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.
- [CW]C. K. Chui, J. Z. Wang (1992):A general framework for compactly supported splines and wavelets. J. Approx. Theory,71:263–304.
- [CW1]C. K. Chui, J. Z. Wang (1992):On compactly supported spline wavelets and a duality principle. Trans. Amer. Math. Soc.,330:903–912.
- [CSW]C. K. Chui, J. Stöckler, J. D. Ward (1992):Compactly supported box spline wavelets. Approx. Theory & Appl.,8:77–100.
- [D]I. Daubechies (1988):Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math.,XLI:909–996.
- [D1]I. Daubechies (1990):The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory,36:961–1005.
- [DR]N. Dyn, A. Ron (1990):Local approximation by certain spaces of multivariate exponential-polynomials, approximation order of exponential box splines and related interpolation problems. Trans. Amer. Math. Soc.,319:381–404.
- [JM]R. Q. Jia, C. A. Micchelli (1991):Using the refinement equation for the construction of pre-wavelets, II: powers of two. In: Curves and Surfaces (P. J. Laurent, A. Le Méhauté, L. L. Schumaker, eds.), New York: Academic Press, 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.
- [Ma]S. G. Mallat (1989):Multiresolution approximations and wavelet orthonormal bases of L 2(R). Trans. Amer. Math. Soc.,315:69–87.
- [Me]Y. Meyer (1980): Ondelettes et Opérateurs I: Ondelettes. Paris: Hermann.
- [Mi]C. A. Micchelli (1991):Using the refinement equation for the construction of pre-wavelets. Numer. Algorithms,1:75–116.
- [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.
- [RS]S. D. Riemenschneider, Z. Shen (1991):Box splines, cardinal series, and wavelets. In: Approximation Theory and Functional Analysis (C. K. Chui, ed.). New York: Academic Press, pp. 133–149.
- [RS1]S. D. Riemenschneider, Z. Shen (1992):Wavelets and prewavelets in low dimensions. J. Approx. Theory,71:18–38.
- [R1]A. Ron (1988):Exponential box splines. Constr. Approx.,4:357–378.
- [R2]A. Ron (1990):Factorization theorems of univariate splines on regular grids. Israel J. Math.,70:48–68.
- [Ru]W. Rudin (1974): Rela, and Complex Analysis. New York: McGraw-Hill.
- [Sö]J. Stöckler (1992):Multivariate wavelets. In: Wavelets—A Tutorial in Theory and Applications (C. K. Chui ed.). New York: Academic Press, 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.