The previous definition of the scaling with the coefficient $\alpha$ lead to the Hermite polynomials
and Hermite functions
So, when $\alpha$ is less than 1, the function is shrinking towards the y-axis which made perfect sense (smaller alpha, smaller function).
This was okay until I realised that this was their definition in the frequency and not the time domain.
Now, this is where the weird effects happen. When $\alpha$ is small in the frequency domain, the Hermite functions in the time domain are getting stretched. With this unintuitive behaviour, it's getting super confusing to select $\alpha$ properly given only the time domain.
Therefore - now that there is if at all a handful of people using this repository - I need to change the definition to $\alpha{new}=\frac{1}{\alpha{old}}$ which leads to
The previous definition of the scaling with the coefficient $\alpha$ lead to the Hermite polynomials
and Hermite functions
So, when $\alpha$ is less than 1, the function is shrinking towards the y-axis which made perfect sense (smaller alpha, smaller function).
This was okay until I realised that this was their definition in the frequency and not the time domain. Now, this is where the weird effects happen. When $\alpha$ is small in the frequency domain, the Hermite functions in the time domain are getting stretched. With this unintuitive behaviour, it's getting super confusing to select $\alpha$ properly given only the time domain. Therefore - now that there is if at all a handful of people using this repository - I need to change the definition to $\alpha{new}=\frac{1}{\alpha{old}}$ which leads to
and