 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 $L^2(\mathbb{R})={⨁}_{j\in \mathbb{Z}}{W}_j$, 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 isEquivalently, with the refinement coefficients from the scaling refinement equation, put $g_k=(-1{)}^{k-1}\overline{h_{1-k}}$ and $\psi (x)=\sqrt{2}{\sum }_k{g}_k\phi (2x-k)$. 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 ${\psi }_{j,k}(x)=2^{j/2}\psi (2^{j}x-k)$ for $j,k\in \mathbb{Z}$.