Scaling function and refinement equation

Remarks

Orthonormality of the translates of $\varphi$ gives $\sum_k h_k \overline{h_{k + 2m}} = \delta_{m,0}$ for all $m \in \Z$. If $\varphi$ is integrable with $\int \varphi \ne 0$, integrating the refinement equation gives $\sum_k h_k = \sqrt2$. The function $\psi(x) = \sqrt2 \sum_k (-1)^k \overline{h_{1-k}} \, \varphi(2x - k)$ is then an orthonormal wavelet.

Example: in the Haar multiresolution analysis, $\varphi = 1_{[0, 1)}$ and $\varphi(x) = \varphi(2x) + \varphi(2x - 1)$. So $h_0 = h_1 = 1/\sqrt2$ and $h_k = 0$ otherwise. The formula for $\psi$ gives $\psi(x) = \varphi(2x) - \varphi(2x - 1) = 1_{[0, 1/2)}(x) - 1_{[1/2, 1)}(x)$, the Haar wavelet.

Non-example: $\varphi = 1_{[0, 2)}/\sqrt2$ satisfies the refinement equation $\varphi(x) = \varphi(2x) + \varphi(2x - 2)$, but its integer translates overlap, so $\ip{\varphi}{\varphi(\cdot - 1)} = \tfrac12 \ne 0$ and it is not the scaling function of a multiresolution analysis in the sense above.