Jun Fang CHENGDeng Feng LI
Let $E=\binom{1\ \ 1}{1 \ -1}$ or $\binom{ \,0 \ \,2\,}{ \,1 \ \,0\,}$, $\psi(x)\in
L^2(R^2)\ \mbox{and}\ \psi_{jk}(x)=2^{\frac{j}{2}}\psi(E^jx -k),$
where $ j\in Z,\, k\in Z^2.$ $\psi(x)$ is called an $E$-tight
frame wavelet if $\{\psi_{jk}\,|\,j\in Z,\, k\in Z^2\}$ is a tight
frame for $L^2(R^2)$. In this paper, a sufficient and necessary
condition for an $E$-tight frame wavelet to be an MRA $E$-tight
frame wavelet is presented. Precisely, an $E$-tight frame wavelet
$\psi$ is an MRA $E$-tight frame wavelet if and only if the
dimension of a particular linear space $F_\psi(\xi)$ is either $0$
or $1$, where $F_\psi(\xi)=\overline{\rm
span}\{\Psi_j(\xi)\,|\,j\geq 1\},\,
\Psi_j(\xi)=\{\hat{\psi}({(E^T)}^j(\xi+2k\pi))\}_{k\in Z^2},\,
j\geq 1.$