Proceedings of the American Mathematical Society

Author(s): Bin Dong; Zuowei Shen
Journal: Proc. Amer. Math. Soc. 134 (2006), 2685-2694.
MSC (2000): Primary 42C40, 41A30
Posted: March 23, 2006
Retrieve article in: PDF DVI PostScript

Abstract: In this paper, we show that the shifts of a pseudo-spline are linearly independent. This is stronger than the (more obvious) statement that the shifts of a pseudo-spline form a Riesz system. In fact, the linear independence of a compactly supported (refinable) function and its shifts has been studied in several areas of approximation and wavelet theory. Furthermore, the linear independence of the shifts of a pseudo-spline is a necessary and sufficient condition for the existence of a compactly supported function whose shifts form a biorthogonal dual system of the shifts of the pseudo-spline.

1.: C. de Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978. MR 0507062 (80a:65027)
2.: J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th ed., McGraw-Hill Higher Education, Boston, 2004. MR 0730937 (84k:30002)
3.: C. de Boor, R. DeVore and A. Ron, The structure of finitely generated shift-invariant spaces in $L_2(\mathbb{R}^d)$ , J. Funct. Anal. 119 (1994), 37-78. MR 1255273 (95g:46050)
4.: A. Ben-Artzi and A. Ron, On the integer translates of a compactly supported function: dual bases and linear projectors, SIAM J. Math. Anal. 21 (1990), 1550-1562. MR 1075591 (91j:41009)
5.: A. Cohen and I. Daubechies, A stability criterion for biorthogonal wavelet bases and their related subband coding scheme, Duke Math. J. 68 (2) (1992), 313-335. MR 1191564 (94b:94005)
6.: A. Cohen, I. Daubechies and J. C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), 485-560. MR 1162365 (93e:42044)
7.: I. Daubechies, Ten Lectures on Wavelets, in: CBMS Conf. Series in Appl. Math., vol. 61, SIAM, Philadelphia, 1992. MR 1162107 (93e:42045)
8.: I. Daubechies, B. Han, A. Ron, and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003), 1-46. MR 1971300 (2004a:42046)
9.: W. Dahmen and C. A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52 (1983), 217-234. MR 0709352 (85e:41033)
10.: W. Dahmen and C. A. Micchelli, On the local linear independence of translates of a box spline, Studia Math. 82 (1985), 243-262. MR 0825481 (87k:41008)
11.: B. Dong and Z. Shen, Pseudo-splines, wavelets and framelets, preprint (2004).
12.: S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986), 185-204. MR 0829123 (88b:41003)
13.: R. Q. Jia, Linear independence of translates of a box spline, J. Approx. Theory 40 (1984), 158-160. MR 0732698 (85h:41025)
14.: R. Q. Jia, Local linear independence of the translates of a box spline, Constr. Approx. 1 (1985), 175-182. MR 0891538 (88d:41017)
15.: R. Q. Jia and C. A. Micchelli, Using the refinement equations for the construction of pre-wavelets. II. Powers of two, Curves and Surfaces, 209-246, Academic Press, Boston, MA, 1991. MR 1123739 (93e:65024)
16.: R. Q. Jia and J. Z. Wang, Stability and linear independence associated with wavelet decompositions, Proc. Am. Math. Soc. 117 (4) (1993), 1115-1124. MR 1120507 (93e:42046)
17.: P. G. Lemarié-Rieusset, On the existence of compactly supported dual wavelets, Appl. Comput. Harmon. Anal. 3 (1997), 117-118. MR 1429683 (97h:42018)
18.: P. G. Lemarié-Rieusset, Fonctions d'échelle interpolantes, polynômes de Bernstein et ondelettes non stationnaires, Rev. Mat. Iberoamericana 13 (1) (1997), 91-188. MR 1462330 (98k:42045)
19.: A. Ron, A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution, Constr. Approx. 5 (1989), 297-308. MR 0996932 (90g:41019)
20.: A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of $L_2(\mathbb{R}^d)$ , Canad. Math. J. 47 (1995), 1051-1094. MR 1350650 (96k:42049)
21.: A. Ron and Z. Shen, Affine systems in $L_2({\mathbb{R}}^d)$ : the analysis of the analysis operator, J. Funct. Anal. 148 (2) (1997), 408-447. MR 1469348 (99g:42043)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42C40, 41A30

Bin Dong
Affiliation: Department of Mathematics, National University of Singapore, Science Drive 2, Singapore 117543, Singapore
Address at time of publication: Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, California 90095-1555
Email: g0301173@nus.edu.sg; bdong@math.ucla.edu

Zuowei Shen
Affiliation: Department of Mathematics, National University of Singapore, Science Drive 2, Singapore 117543, Singapore
Email: matzuows@nus.edu.sg

DOI: 10.1090/S0002-9939-06-08316-X
PII: S 0002-9939(06)08316-X
Keywords: Linear independence, pseudo-spline, stability.
Received by editor(s): September 22, 2004
Received by editor(s) in revised form: April 6, 2005
Posted: March 23, 2006
Additional Notes: This research was supported by several grants from the Department of Mathematics, National University of Singapore.
Communicated by: David R. Larson
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS