Wavelet

Remarks

The factor $2^{j/2}$ makes $\norm{\psi_{j,k}}_2 = \norm{\psi}_2$. Then every $f \in L^2(\R)$ has the expansion $f = \sum_{j,k} \ip{f}{\psi_{j,k}} \psi_{j,k}$, convergent in $L^2(\R)$. If $\psi$ is also integrable, then $\int_\R \psi(x) \, dx = 0$. Weaker notions are also called wavelets: $\psi$ such that $(\psi_{j,k})$ is a Riesz basis or a frame, and the admissible functions of the continuous wavelet transform.

Example: the Haar wavelet $\psi = 1_{[0, 1/2)} - 1_{[1/2, 1)}$ is an orthonormal wavelet. For instance, $\psi_{0,0}$ and $\psi_{1,0}$ are orthogonal, because $\psi_{1,0}$ lives on $[0, \tfrac12)$, where $\psi_{0,0} = 1$, and has integral $0$.

Non-example: the Gaussian $\psi(x) = e^{-\pi x^2}$ is not an orthonormal wavelet. Its translates $\psi_{0,0}$ and $\psi_{0,1}$ are both positive, so $\ip{\psi_{0,0}}{\psi_{0,1}} > 0$ and they are not orthogonal.