I’m not a mathematician and I’m working with some transforms in physical chemistry.
I use a transform to pass from the time domain of phase domain in a process that use a square wave to perturb and modulate a stimulus on a system. In this case, the transform is $$ A(\phi)= \int_{0}^{T} A(t)\sin(kwt + \phi)\,dt, $$ where
- $k$ is the frequent of the modulation,
- $w$ is the frequency,
- $t$ is the time and
- $\phi$ is the phase.
After the stimulation and the definition of $A(\phi)$ I want to pass again from the phase domain to the time domain, so to (re)calculate $A(t)$ in the range between $0$ and $2\pi$. So, what is the inverse kernel of $\sin(kwt + \phi)$? Is it the same?
Just for your information, $k$ can only be a odd integer.
I hope you can help me.
Thank you in advance!