Please use this identifier to cite or link to this item:
|Title:||Pseudo-splines, wavelets and framelets|
|Source:||Dong, B., Shen, Z. (2007-01). Pseudo-splines, wavelets and framelets. Applied and Computational Harmonic Analysis 22 (1) : 78-104. ScholarBank@NUS Repository. https://doi.org/10.1016/j.acha.2006.04.008|
|Abstract:||The first type of pseudo-splines were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1-46; I. Selesnick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2) (2001) 163-181] to construct tight framelets with desired approximation orders via the unitary extension principle of [A. Ron, Z. Shen, Affine systems in L2 (Rd): The analysis of the analysis operator, J. Funct. Anal. 148 (2) (1997) 408-447]. In the spirit of the first type of pseudo-splines, we introduce here a new type (the second type) of pseudo-splines to construct symmetric or antisymmetric tight framelets with desired approximation orders. Pseudo-splines provide a rich family of refinable functions. B-splines are one of the special classes of pseudo-splines; orthogonal refinable functions (whose shifts form an orthonormal system given in [I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909-996]) are another class of pseudo-splines; and so are the interpolatory refinable functions (which are the Lagrange interpolatory functions at Z and were first discussed in [S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986) 185-204]). The other pseudo-splines with various orders fill in the gaps between the B-splines and orthogonal refinable functions for the first type and between B-splines and interpolatory refinable functions for the second type. This gives a wide range of choices of refinable functions that meets various demands for balancing the approximation power, the length of the support, and the regularity in applications. This paper will give a regularity analysis of pseudo-splines of the both types and provide various constructions of wavelets and framelets. It is easy to see that the regularity of the first type of pseudo-splines is between B-spline and orthogonal refinable function of the same order. However, there is no precise regularity estimate for pseudo-splines in general. In this paper, an optimal estimate of the decay of the Fourier transform of the pseudo-splines is given. The regularity of pseudo-splines can then be deduced and hence, the regularity of the corresponding wavelets and framelets. The asymptotical regularity analysis, as the order of the pseudo-splines goes to infinity, is also provided. Furthermore, we show that in all tight frame systems constructed from pseudo-splines by methods provided both in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1-46] and this paper, there is one tight framelet from the generating set of the tight frame system whose dilations and shifts already form a Riesz basis for L2 (R). © 2006 Elsevier Inc. All rights reserved.|
|Source Title:||Applied and Computational Harmonic Analysis|
|Appears in Collections:||Staff Publications|
Show full item record
Files in This Item:
There are no files associated with this item.
checked on Apr 10, 2018
WEB OF SCIENCETM
checked on Apr 10, 2018
checked on Mar 12, 2018
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.