Continuous wavelet transform

Remarks

The integral converges by the Cauchy–Schwarz inequality, and $\abs{W_\psi f(a, b)} \le \norm{f}_2 \norm{\psi}_2$. Taking the Fourier transform in $b$ and using the Plancherel theorem gives Calderón's identity $$\int_{\R \setminus \set{0}} \int_\R \abs{W_\psi f(a, b)}^2 \, db \, \frac{da}{a^2} = C_\psi \norm{f}_2^2.$$ So $f$ can be recovered from $W_\psi f$, in a weak sense, by $f = C_\psi^{-1} \int \int W_\psi f(a, b) \, \abs{a}^{-1/2} \psi((\cdot - b)/a) \, db \, da / a^2$. Many authors take only $a > 0$; then the constant changes and, for real $\psi$, one obtains $\tfrac12 C_\psi$.

Example: the Mexican hat $\psi(t) = (1 - t^2) e^{-t^2/2}$ has $\hat\psi(\xi)$ equal to a constant times $\xi^2 e^{-2\pi^2 \xi^2}$, so $C_\psi < \infty$ and $W_\psi$ is defined.

Non-example: for the Gaussian $\psi(t) = e^{-\pi t^2}$, $\hat\psi(0) = 1$, and $C_\psi$ diverges near $\xi = 0$. The formula for $W_\psi f$ still makes sense, but Calderón's identity fails, and this choice is excluded by the definition.