On compactly supported spline wavelets and a duality principle
HTML articles powered by AMS MathViewer
- by Charles K. Chui and Jian-zhong Wang
- Trans. Amer. Math. Soc. 330 (1992), 903-915
- DOI: https://doi.org/10.1090/S0002-9947-1992-1076613-3
- PDF | Request permission
Abstract:
Let $\cdots \subset {V_{ - 1}} \subset {V_0} \subset {V_1} \subset \cdots$ be a multiresolution analysis of ${L^2}$ generated by the $m$th order $B$-spline ${N_m}(x)$. In this paper, we exhibit a compactly supported basic wavelet ${\psi _m}(x)$ that generates the corresponding orthogonal complementary wavelet subspaces $\cdots ,{W_{ - 1}},{W_0},{W_1}, \ldots$. Consequently, the two finite sequences that describe the two-scale relations of ${N_m}(x)$ and ${\psi _m}(x)$ in terms of ${N_m}(2x - j),j \in \mathbb {Z}$, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases $\{ {\tilde N_m}(x - j)\}$ and $\{ {\tilde \psi _m}(x - j)\}$, relative to $\{ {N_m}(x - j)\}$ and $\{ {\psi _m}(x - j)\}$, respectively.References
- Charles K. Chui, Multivariate splines, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by Harvey Diamond. MR 1033490, DOI 10.1137/1.9781611970173
- Charles K. Chui and Jian-zhong Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), no. 3, 785–793. MR 1077784, DOI 10.1090/S0002-9939-1991-1077784-X
- Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909–996. MR 951745, DOI 10.1002/cpa.3160410705
- Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases of $L^2(\textbf {R})$, Trans. Amer. Math. Soc. 315 (1989), no. 1, 69–87. MR 1008470, DOI 10.1090/S0002-9947-1989-1008470-5
- I. J. Schoenberg, Cardinal spline interpolation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0420078
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 330 (1992), 903-915
- MSC: Primary 41A15; Secondary 41A05, 41A30, 42C05
- DOI: https://doi.org/10.1090/S0002-9947-1992-1076613-3
- MathSciNet review: 1076613