Local dual and poly-scale refinability

Sun, Qiyu

doi:10.1090/S0002-9939-04-07622-1

Local dual and poly-scale refinability
HTML articles powered by AMS MathViewer

by Qiyu Sun PDF

Proc. Amer. Math. Soc. 133 (2005), 1175-1184 Request permission

Abstract:

For a compactly supported function $f$, let $S_n(f), n\ge 0$, be the space of all finite linear combinations of $f(M^n\cdot -k), k\in \mathbf Z$. In this paper, we consider the explicit construction of local duals of $f$ and the poly-scale refinability of functions in $S_0(f)$ when $f$ is an $M$-refinable function. We show that for any $M$-refinable function $f$, there exists a local dual of $f$ in $S_N(f)$ for some $N\ge 0$, and that any function in $S_0(f)$ is poly-scale refinable.

References

Asher Ben-Artzi and Amos Ron, On the integer translates of a compactly supported function: dual bases and linear projectors, SIAM J. Math. Anal. 21 (1990), no. 6, 1550–1562. MR 1075591, DOI 10.1137/0521085
Charles K. Chui (ed.), Wavelets, Wavelet Analysis and its Applications, vol. 2, Academic Press, Inc., Boston, MA, 1992. A tutorial in theory and applications. MR 1161244, DOI 10.1016/B978-0-12-174590-5.50029-0
Charles K. Chui, Wenjie He, and Joachim Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal. 13 (2002), no. 3, 224–262. MR 1942743, DOI 10.1016/S1063-5203(02)00510-9
Charles K. Chui, Wenjie He, Joachim Stöckler, and Qiyu Sun, Compactly supported tight affine frames with integer dilations and maximum vanishing moments, Adv. Comput. Math. 18 (2003), no. 2-4, 159–187. Frames. MR 1968118, DOI 10.1023/A:1021318804341
Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
Ingrid Daubechies, Bin Han, Amos Ron, and Zuowei Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003), no. 1, 1–46. MR 1971300, DOI 10.1016/S1063-5203(02)00511-0
S. Dekel and N. Dyn, Poly-scale refinability and subdivision, Appl. Comput. Harmon. Anal. 13 (2002), no. 1, 35–62. MR 1930175, DOI 10.1016/S1063-5203(02)00006-4
George C. Donovan, Jeffrey S. Geronimo, and Douglas P. Hardin, Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets, SIAM J. Math. Anal. 27 (1996), no. 6, 1791–1815. MR 1416519, DOI 10.1137/S0036141094276160
P. G. Lemarié-Rieusset, On the existence of compactly supported dual wavelets, Appl. Comput. Harmon. Anal. 4 (1997), no. 1, 117–118. MR 1429683, DOI 10.1006/acha.1996.0199
Rong Qing Jia and Charles A. Micchelli, On linear independence for integer translates of a finite number of functions, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 69–85. MR 1200188, DOI 10.1017/S0013091500005903
Qi Yu Sun, Compactly supported distributional solutions of nonstationary nonhomogeneous refinement equations, Acta Math. Sin. (Engl. Ser.) 17 (2001), no. 1, 1–14. MR 1831741, DOI 10.1007/s101140000092
Q. Sun, N. Bi and D. Huang, “An Introduction to Multiband Wavelets”, Zhejiang University Press, China, 2001.
Amos Ron, A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution, Constr. Approx. 5 (1989), no. 3, 297–308. MR 996932, DOI 10.1007/BF01889611
Kang Zhao, Global linear independence and finitely supported dual basis, SIAM J. Math. Anal. 23 (1992), no. 5, 1352–1355. MR 1177795, DOI 10.1137/0523077