= Solution
An orthonormal <multiresolution analysis> consists of closed subspaces $V_j\subset L_2(\mathbb R)$, $j\in\mathbb Z$, with $V_j\subset V_{j+1}$, $\overline{\bigcup_jV_j}=L_2(\mathbb R)$, $\bigcap_jV_j=\{0\}$, and the dilation equivalence $h\in V_j\iff h(2\cdot)\in V_{j+1}$. A <scaling function> $\phi$ has integer translates forming an <orthonormal basis> of $V_0$. Hence
$$
\phi_{j,k}(x)=2^{j/2}\phi(2^jx-k),\qquad k\in\mathbb Z,
$$
is an <orthonormal basis> of $V_j$. Set $W_j=V_{j+1}\ominus V_j$, the <orthogonal complement> of $V_j$ inside $V_{j+1}$. A function $\psi$ whose integer translates form an <orthonormal basis> of $W_0$ produces a complete <orthonormal wavelet> system $\{2^{j/2}\psi(2^j\cdot-k):j,k\in\mathbb Z\}$: different $W_j$ are orthogonal, their sum contains every difference $V_b\ominus V_a$, and density together with the zero intersection exhausts $L_2(\mathbb R)$.
Here is the filter construction that guarantees such a <wavelet>. Nesting puts $\phi$ in $V_1$, so expansion in its <orthonormal basis> gives the <scaling refinement equation> $\phi=\sum_na_n\phi(2\cdot-n)$, with $a\in\ell_2$. Define the <MRA low-pass filter> and its complementary <quadrature mirror filter> by
$$
m(t)=\tfrac12\sum_na_ne^{-int},\qquad q(t)=-e^{-it}\overline{m(t+\pi)}.
$$
Splitting the <orthonormal translates and Fourier periodization> identity at $2t$, as in part (b), gives $|m(t)|^2+|m(t+\pi)|^2=1$ almost everywhere. Consequently the rows of
$$
\begin{pmatrix}m(t)&m(t+\pi)\\-e^{-it}\overline{m(t+\pi)}&e^{-it}\overline{m(t)}\end{pmatrix}
$$
have length one and are orthogonal; the matrix is unitary. Put $b_n=(-1)^n\overline{a_{1-n}}$ and $\psi=\sum_nb_n\phi(2\cdot-n)$, so $q(t)=\tfrac12\sum_nb_ne^{-int}$.
To see completeness as well as orthogonality, identify $V_1$ with its fine-scale coefficient sequences and take their <Fourier series>. Group the frequencies $t$ and $t+\pi$ over a half-period. The two rows of the displayed matrix are the normalized coarse and complementary channels; their unitarity is exactly an onto isometry from these two channels to the fine-scale coefficient space. Translates of $\phi$ and $\psi$ correspond to the <Fourier basis> in the respective channels. They therefore jointly form an <orthonormal basis> of $V_1$. The <orthogonal complement> channel gives the <wavelet completion of a multiresolution filter>, namely the integer translates of $\psi$ span $W_0$. This proves the asserted relation with an <orthonormal wavelet>.
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