An application of nonstationary wavelets
© Zhao; licensee Springer. 2014
Received: 25 December 2013
Accepted: 16 April 2014
Published: 2 May 2014
Using nonstationary wavelets, we investigate the wavelet expansion in the standard Besov spaces. Especially, the nonstationary wavelets’ characterization for Besov spaces is given.
MSC:42C40, 35Q30, 41A15.
In this paper, we shall study the nonstationary wavelet  expansion in the standard Besov spaces. Nonstationary wavelet systems are generally obtained from a sequence of nonstationary refinable functions.
where are 2π-periodic measurable functions, called refinement masks, or simply masks.
where are 2π-periodic measurable functions called wavelet masks. The masks and satisfy . When , let , and , where . Define . Then the following theorem holds.
Theorem 1.1 (i) (Theorem 1.3, ). Let (mask for the pseudo-spline of type I with order ). Then is a compactly supported and wavelet frame in for arbitrary ;
(ii) (Theorem 1.4, ). When , then () is a compactly supported orthonormal basis in , and () has vanishing moments. More precisely, and with uniform constant .
Here, with and denotes the M th order smooth modulus of a function f, defined by as usual. The classical difference operator is defined by , as well as for a positive integer . The Besov (quasi-)norm is given by and the two integers yield equivalent norms (, Remark 3.2.2).
for , . However, due to the regularity restrictions of the Hermite splines, their characterization requires in the quadratic case and in the cubic one (e.g. [5, 7, 8]). In , we remove that restriction of s by using the B-spline wavelets with weak duals as introduced in , but the supports of the wavelets become larger as s increases. So, the main result of this paper is to characterize Besov spaces via nonstationary wavelets because of their arbitrary smoothness and uniform support.
for , with Lebesgue measurable set , , and . For a Lebesgue measurable function f, the support of f means the set , which is well defined up to a set of measure 0. Define throughout this paper.
Now, we state the Main Theorem of this paper.
for , .
2 Proof of Main Theorem
This section is devoted to proving the Main Theorem. We begin with three lemmas for proving upper and lower bounds of the characterization.
Lemma 2.1 Let , , be arbitrary, then and with a uniform constant for .
for leads to for all . holds similarly.
as well as for with a uniform constant . Thus, the result holds. □
Note that the constants are uniform because of .
where and with .
Now, we are in the position to show the Main Theorem.
with , , , , , and .
Remark 2.2 Some questions are left to be considered. Note that we assume in our Main Theorem. Then a natural question is to study the case for . Another one is to discuss whether or not the wavelet frames of (i) in Theorem 1.1 can characterize Besov spaces. The last question is to relax the restriction .
The author would like to thank the referees for their helpful comments. This work is supported by the National Natural Science Foundation of China, Tian Yuan Foundation (Grant No. 11226106) and the National Natural Science Foundation of China (Grant No. 11201094).
- Han B, Shen Z: Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames. Isr. J. Math. 2009, 172: 371-398. 10.1007/s11856-009-0079-9MathSciNetView ArticleGoogle Scholar
- Han B, Shen Z:Compactly supported symmetric wavelets with spectral approximation order. SIAM J. Math. Anal. 2009, 40: 905-938.MathSciNetView ArticleGoogle Scholar
- Hans T: Theory of Function Spaces II. Birkhäuser, Basel; 1992.Google Scholar
- Härdle W, Kerkyacharian G, Picard D, Tsybakov A: Wavelets, Approximation and Statistal Applications. Springer, Berlin; 1998.View ArticleGoogle Scholar
- Bittner K, Urban K: On interpolatory divergence-free wavelets. Math. Comput. 2007,76(258):903-929. 10.1090/S0025-5718-06-01949-1MathSciNetView ArticleGoogle Scholar
- Cohen A: Wavelet methods in numerical analysis. Handb. Numer. Anal. VII. In Handbook of Numerical Analysis. North-Holland, Amsterdam; 2000:417-711.Google Scholar
- Liu Y, Zhao J: Convergence of Hermite interpolatory operators. Sci. China Math. 2010,53(8):2115-2126. 10.1007/s11425-010-4047-yMathSciNetView ArticleGoogle Scholar
- Zhao J: Interpolatory Hermite splines on rectangular domains. Appl. Math. Comput. 2010, 216: 2799-2813. 10.1016/j.amc.2010.03.130MathSciNetView ArticleGoogle Scholar
- Liu Y, Zhao J: An extension of Bittner and Urban’s theorem. Math. Comput. 2013,82(281):401-411.View ArticleGoogle Scholar
- Jia RQ, Wang JZ, Zhou DX: Compactly supported wavelets bases for Sobolev spaces. Appl. Comput. Harmon. Anal. 2003, 15: 224-241. 10.1016/j.acha.2003.08.003MathSciNetView ArticleGoogle Scholar
- Cohen A, Dyn N: Nonstationary subdivision schemes and multiresolution analysis. SIAM J. Math. Anal. 1996, 27: 1745-1769. 10.1137/S003614109427429XMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.