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

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Bin Han and Qun Mo

## Abstract

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.

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