设 $E=(\genfrac{}{}{0ex}{}{11}{1-1})$ 或 $(\genfrac{}{}{0ex}{}{02}{10})$, $\psi (x)\in L^2(R^2)\text{且}{\psi }_{jk}(x)=2^{\frac{j}{2}}\psi (E^{j}x-k),$ 其中 $j\in Z,k\in Z^2.$ 若 $\{{\psi }_{jk}|j\in Z,k\in Z^2\}$ 构成 $L^2(R^2)$ 的紧框架, 则称 $\psi (x)$ 为 $E$-紧框架小波. 本文给出$E$-紧框架小波是 MRA $E$-紧框架小波 的一个充要条件, 即 $E$ 紧框架小波 $\psi$ 来自多尺度分析当且仅当 线性空间 $F_{\psi }(\xi )$ 的维数为 $0$ 或 $1$, 其中 $F_{\psi }(\xi )=\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}\{{\mathrm{\Psi }}_j(\xi )|j\ge 1\},{\mathrm{\Psi }}_j(\xi )=\{\hat{\psi }({(E^T)}^j(\xi +2k\pi )){\}}_{k\in Z^2},j\ge 1.$

Let $E=(\genfrac{}{}{0ex}{}{11}{1-1})$ or $(\genfrac{}{}{0ex}{}{02}{10})$, $\psi (x)\in L^2(R^2)\text{and}{\psi }_{jk}(x)=2^{\frac{j}{2}}\psi (E^{j}x-k),$ 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 $F_{\psi }(\xi )=\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}\{{\mathrm{\Psi }}_j(\xi )|j\ge 1\},{\mathrm{\Psi }}_j(\xi )=\{\hat{\psi }({(E^T)}^j(\xi +2k\pi )){\}}_{k\in Z^2},j\ge 1.$