Compactly supported wavelet bases for Sobolev spaces

Jia, Rong-Qing; Jian-zhong, Wang; Zhou, Ding‐Xuan

doi:10.1016/j.acha.2003.08.003

Cited by 56 publications

(39 citation statements)

References 19 publications

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“…There are two interesting fundamental questions about MRA Riesz wavelet bases derived from the refinable function φ. Some sufficient conditions for Question A have been given in the literature [2,10,11,13,14]. More recently, it has been shown in [10] and [6] that for all B-spline functions, all refinable interpolating functions with maximum approximation orders, and all pseudo-spline functions (for pseudo-spline functions, see [5]), Question B indeed holds.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On a conjecture about MRA Riesz wavelet bases

Han

2005

Proc. Amer. Math. Soc.

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Abstract. Let φ be a compactly supported refinable function in L 2 (R) such that the shifts of φ are stable andφ(2ξ) =â(ξ)φ(ξ) for a 2π-periodic trigonometric polynomialâ. A wavelet function ψ can be derived from φ byψ(2ξ) := e −iξâ (ξ + π)φ(ξ). If φ is an orthogonal refinable function, then it is well known that ψ generates an orthonormal wavelet basis in L 2 (R). Recently, it has been shown in the literature that if φ is a B-spline or pseudo-spline refinable function, then ψ always generates a Riesz wavelet basis in L 2 (R). It was an open problem whether ψ can always generate a Riesz wavelet basis in L 2 (R) for any compactly supported refinable function in L 2 (R) with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function ψ does not generate a Riesz wavelet basis in L 2 (R). Our proof is based on some necessary and sufficient conditions on the 2π-periodic functionsâ andb in C ∞ (R) such that the wavelet function ψ, defined byψ(2ξ) :=b(ξ)φ(ξ), generates a Riesz wavelet basis in L 2 (R).

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On a conjecture about MRA Riesz wavelet bases

Han

2005

Proc. Amer. Math. Soc.

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“…Hence, there is no need for us to check the stability of φ, which could be a highly nontrivial task. Moreover, it has been shown in [34, Theorem 3.2] (also see [37]), Corollary 1.7 [12,14,16,23,24,33,34,35,39,40,44,47]). …”

Section: Corollary 17 Letâ Andb Be 2π-periodic Functions With Exponmentioning

confidence: 99%

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

Han

Shen

2008

Constr Approx

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Abstract. This paper generalizes the mixed extension principle in L 2 (R For s > 0, the key of this general mixed extension principle is the regularity of , k ∈ Z} is a wavelet frame in H s (R) for any 0 < s < m − 1/2, where B m is the B-spline of order m. This simple construction is also applied to multivariate box splines to obtain wavelet frames with short supports, noting that it is hard to construct nonseparable multivariate wavelet frames with small supports. Applying this general mixed extension principle, we obtain and characterize dual Riesz bases (. For example, all interpolatory wavelet systems in [25] generated by an interpolatory refinable function φ ∈ H s (R) with s > 1/2 are Riesz bases of the Sobolev space H s (R). This general mixed extension principle also naturally leads to a characterization of the Sobolev norm of a function in terms of weighted norm of its wavelet coefficient sequence (decomposition sequence) without requiring that dual wavelet frames should be in L 2 (R d ), which is quite different to other approaches in the literature.

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“…For a comprehensive study on box splines, we refer the reader to the book [1] by de Boor, Höllig, and Riemenschneider. Recently, compactly supported wavelet bases for Sobolev spaces were studied by Lorentz and Oswald [26], and by Jia, Wang, and Zhou [21].…”

Section: Introductionmentioning

confidence: 99%

$C^1$ spline wavelets on triangulations

Jia

Liu

2008

Math. Comp.

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Abstract. In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in C 1 wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of C 1 quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct C 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations. IntroductionIn this paper we investigate spline wavelets on general triangulations. In particular, we are interested in C 1 wavelets generated from piecewise quadratic polynomials.In µ with 1 < µ < 5/2. In this paper we shall employ the Hermite interpolation property of the PowellSabin elements to construct C 1 spline wavelets on general triangulations. These wavelet bases will be shown to be H 2 -stable. In the process we shall establish a general theory for wavelet bases in Besov spaces, which will be useful for future research.Here is an outline of the paper. In Section 2 we review some basic properties of Besov spaces as well as Bernstein type inequalities for spline functions on polygonal domains. In Section 3, using the Powell-Sabin elements, we formulate bases for the concerned spaces of spline functions and establish Jackson type inequalities. In Section 4 we discuss multiresolution analysis of Besov spaces and related norm equivalence. In Section 5 we develop a general theory for wavelet bases in Besov spaces. Finally, in Section 6, we construct C 1 wavelet bases on general triangulations and show that the wavelet bases are H 2 -stable. In particular, for the three-direction mesh, the wavelets are explicitly given.It is expected that the C 1 spline wavelets constructed in this paper will have applications to numerical solutions of partial differential equations. These wavelet bases are particularly suitable for the biharmonic equation (see the discussion in [27] and [10]). The condition numbers of the corresponding discretization matrices will be uniformly bounded. Our wavelet bases could have applications to a wide range of problems in numerical analysis, such as numerical solutions of integral equations and operator equations, and singular perturbation problems. See the related work of Chen, Micchelli and Xu [2], Liu and Xu [25], and Shen and Lin [32]. The refinable spline functions discussed in this paper also could have applications to computer graphics and multi-level data representation. See the related work of Chui and Jiang [4,5] on surface subdivision schemes.

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Compactly supported wavelet bases for Sobolev spaces

Cited by 56 publications

References 19 publications

On a conjecture about MRA Riesz wavelet bases

On a conjecture about MRA Riesz wavelet bases

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

$C^1$ spline wavelets on triangulations

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