= Solution
Let $W_j=V_{j+1}\ominus V_j$ denote the <orthogonal complement> of $V_j$ inside $V_{j+1}$. The <Meyer-Mallat theorem> constructs a <wavelet> whose <integer> <function translations> form an <orthonormal basis> of $W_0$. Dilation then gives the <orthogonal> decomposition into <orthogonal complements> $L^2(\mathbb R)=\bigoplus_{j\in\mathbb Z}W_j$, so its translates and dilates form an <orthonormal wavelet> basis.
Using the <MRA low-pass filter>, choose the high-pass symbol $n(\xi)=e^{-i\xi}\overline{m(\xi+\pi)}$. The <quadrature mirror filter> identity makes the two analysis channels <orthonormal>. The resulting <Fourier transform> formula is
$$
\boxed{\widehat\psi(2\xi)=e^{-i\xi}\overline{m(\xi+\pi)}\widehat\varphi(\xi).}
$$
Equivalently, with the refinement coefficients from the <scaling refinement equation>, put $g_k=(-1)^{k-1}\overline{h_{1-k}}$ and $\psi(x)=\sqrt2\sum_kg_k\varphi(2x-k)$. This index choice gives precisely the displayed high-pass symbol. Changing all $g_k$ by one constant unit phase gives the same <orthonormal wavelet> construction. In particular the often-used $(-1)^k\overline{h_{1-k}}$ convention differs only by a global minus sign. The basis is $\psi_{j,k}(x)=2^{j/2}\psi(2^jx-k)$ for $j,k\in\mathbb Z$.
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