= Solution
Use the <Fourier transform> convention $\widehat f(\xi)=\int_{\mathbb R}f(x)e^{-ix\xi}\,dx$. An <orthonormal> <multiresolution analysis> is a family of closed <vector subspaces> $V_j\subset L^2(\mathbb R)$, indexed by <integers>, with $V_j\subset V_{j+1}$, $\overline{\bigcup_jV_j}=L^2(\mathbb R)$, and $\bigcap_jV_j=\{0\}$. Its dilation condition is $f\in V_j$ if and only if $f(2\mathord\cdot)\in V_{j+1}$. The space $V_0$ is invariant under <integer> <function translations> and admits a <scaling function> $\varphi$ whose <integer> <function translations> form an <orthonormal basis>. Consequently $\varphi_{j,k}=2^{j/2}\varphi(2^j\mathord\cdot-k)$ form an <orthonormal basis> of $V_j$.
Since $\varphi\in V_1$, expansion in that <orthonormal basis> gives the <scaling refinement equation>
$$
\varphi(x)=\sqrt2\sum_{k\in\mathbb Z}h_k\varphi(2x-k),\qquad h_k=\langle\varphi,\varphi_{1,k}\rangle.
$$
The <series> converges in the <L2 norm>. The associated <low-pass filter of a multiresolution analysis> is the periodic <Fourier series> symbol
$$
\boxed{m(\xi)=\frac1{\sqrt2}\sum_{k\in\mathbb Z}h_ke^{-ik\xi},\qquad\widehat\varphi(2\xi)=m(\xi)\widehat\varphi(\xi).}
$$
Initially the symbol is defined almost everywhere. <Orthonormal> <integer> <function translations> give the <quadrature mirror filter> identity $|m(\xi)|^2+|m(\xi+\pi)|^2=1$ almost everywhere. For a <Lebesgue integrable> <scaling function>, its <Fourier transform> is <continuous> and $|\widehat\varphi(0)|=1$; choosing a constant phase makes $\widehat\varphi(0)=1$ and the <continuous> representative near zero has $m(0)=1$. Different conventions absorb $\sqrt2$ into the refinement coefficients; the displayed convention fixes that ambiguity.
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