= Solution
For the <multiresolution analysis> $V_j$ and its detail spaces $W_j$, take the completed <tensor product> $\mathcal V_j=V_j\otimes V_j$. Since $V_{j+1}=V_j\oplus W_j$,
$$
\mathcal V_{j+1}=\mathcal V_j\oplus(W_j\otimes V_j)\oplus(V_j\otimes W_j)\oplus(W_j\otimes W_j).
$$
The three <tensor-product wavelets> are
$$
\Psi^1(x,y)=\psi(x)\varphi(y),\quad
\Psi^2(x,y)=\varphi(x)\psi(y),\quad
\Psi^3(x,y)=\psi(x)\psi(y).
$$
Taking products of members of the two one-dimensional families gives an <orthonormal basis> of each respective completed <tensor product>. Density and the coarse-scale trivial intersection yield
$$
\boxed{\{2^j\Psi^a(2^jx-k_1,2^jy-k_2):a=1,2,3,\ j,k_1,k_2\in\mathbb Z\}}
$$
as an <orthonormal basis> of $L^2(\mathbb R^2)$. The normalization is $2^j$ in two dimensions. On the square, tensorize the <interval-adapted wavelet basis> and include the finite coarse <scaling function> part.
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