Symmetric MRA Tight Wavelet Farmes with Three Generators and High Vanishing Moments

Bin Han and Qun Mo


Let $\phi$ be a compactly supported symmetric refinable function in $L_2(\RR)$ with a finitely supported symmetric mask on $\ZZ$. Under the assumption that the shifts of $\phi$ are stable, in this paper we prove that one can always construct three wavelet functions $\psi^1$, $\psi^2$ and $\psi^3$ such that \begin{enumerate} \item[{(i)}] All the wavelet functions $\psi^1$, $\psi^2$ and $\psi^3$ are compactly supported and are finite linear combinations of the functions $\phi(2\cdot-k), k\in \ZZ$; \item[{(ii)}] Each of the wavelet functions $\psi^1$, $\psi^2$ and $\psi^3$ is either symmetric or antisymmetric; \item[{(iii)}] $\{\psi^1, \psi^2, \psi^3\}$ generates a tight wavelet frame in $L_2(\RR)$, that is, $$ \| f\|^2 =\sum_{\ell=1}^3 \sum_{j\in \ZZ} \sum_{k\in \ZZ} |\la f, \psi^\ell_{j,k}\ra|^2 \qquad \forall\; f\in L_2(\RR), $$ where $\psi^\ell_{j,k}:=2^{j/2}\psi^\ell(2^j\cdot-k)$, $\ell=1, 2, 3$ and $j, k\in \ZZ$; \item[{(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 $\phi$. \end{enumerate} %Our result in this paper generalizes several results in the %literature on symmetric tight wavelet frames that are derived from %refinable functions via a multiresolution analysis. We shall give an example to demonstrate that the assumption on stability of the refinable function $\phi$ cannot be dropped. Some examples of symmetric tight wavelet frames with three generators will be given to illustrate the results and construction in this paper.

Back to Preprints and Publications