Symmetric MRA tight wavelet frames with three generators and high vanishing moments. (English) Zbl 1057.42026

Han, Bin; Mo, Qun

Appl. Comput. Harmon. Anal. 18, No. 1, 67-93 (2005).

Summary: Let \(\varphi\) be a compactly supported symmetric real-valued refinable function in \(L_2(\mathbb{R})\) with a finitely supported symmetric real-valued mask on \(\mathbb{Z}\). Under the assumption that the shifts of \(\varphi\) are stable, in this paper we prove that one can always construct three wavelet functions \(\psi^1\), \(\psi^2\), and \(\psi^3\) such that:
(i) All the wavelet functions \(\psi^1\), \(\psi^2\), and \(\psi^3\) are compactly supported, real-valued and finite linear combinations of the functions \(\varphi(2\cdot-k)\), \(k\in\mathbb{Z}\);
(ii) Each of the wavelet functions \(\psi^1\), \(\psi^2\), and \(\psi^3\) is either symmetric or antisymmetric;
(iii) \(\{\psi^1,\psi^2,\psi^3\}\) generates a tight wavelet frame in \(L_2(\mathbb{R})\), that is, \[ \|f\|^2=\sum^3_{\ell=1} \sum_{j \in\mathbb{Z}} \sum_{k\in\mathbb{Z}}\bigl|\langle f,\psi^\ell_{j,k} \rangle \bigr|^2\;\forall f\in L_2(\mathbb{R}), \] where \(\psi^\ell_{j,k}:= 2^{j/2} \psi^\ell(2^j\cdot-k)\), \(\ell=1,2,3\) and \(j,k\in\mathbb{Z}\);
(iv) Each of the wavelet functions \(\psi^1, \psi^2\), and \(\psi^3\) has the highest possible order of vanishing moments, that is, its order of vanishing moments matches the order of the approximation order provided by the refinable function \(\varphi\).
We give an example to demonstrate that the assumption on stability of the refinable function \(\varphi\) cannot be dropped. Some examples of symmetric tight wavelet frames with three compactly supported real-valued symmetric/antisymmetric generators are given to illustrate the results and construction in this paper.

MSC:

42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15	General harmonic expansions, frames
41A15	Spline approximation
41A25	Rate of convergence, degree of approximation

Keywords:

symmetric tight wavelet frames; multiresolution analysis; oblique extension principle; symmetry; vanishing moments; sum rules

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