> I believe the Gabor transform is approximately the same as a continuous wavelet transform using a “frequency 1” Gabor wavelet. I’m not sure.
I no longer believe this! If you compare my formula for the continuous wavelet transform to the formula for the [Gabor transform](https://en.wikipedia.org/wiki/Gabor_transform) you'll see they're very different. And I'm not talking about the fact that the first has a sum while the second has an integral - that's not so important!
I'm talking about the fact that in a continuous wavelet transform we're convolving our function by a rescaled version of a chosen "wavelet", while in the Gabor transform we're multiplying our function by a fixed Gaussian and then taking its Fourier transform. Even if we choose our wavelet to be a Gaussian, these are different things to do.
The continuous wavelet transform sees how much a small segment of our function "matches" a rescaled version of our chosen wavelet $\psi$. So, if $\psi$ has some wiggles, this transform will see how much a small segment of our function has wiggles of a given shape and frequency.