$$s(t) = \cos(wt)\cdot u(t)$$
with $u(t)$ being the unit step.
Suppose we can represent such a signal as the sum of a transient and of a "steady state". A transient is a short-time wide-band spectral component of part of the signal. In this case the transient occurs at $t = 0$.
My question is, given $s(t)$ above, what are $s_t(t)$ and $s_s(t)$ in the decomposition
$$s(t) = s_t(t) + s_s(t)$$
where $s_t(t)$ is localized around t = 0 with minimum support in time and $s_s(t)$ has minimum support in frequency. Essentially we are trying to maximize the time-locality for $s_t(t)$ and maximize the frequency-locality for $s_s(t)$.
Note that solutions such as $s_t(t) = -\cos(wt)\cdot u(-t)$ and $s_s(t) = \cos(wt)$ are not allowed as $s_t(t)$ is not localized around t. In fact I think we should require the transient to be causal in nature. Another possibility is $s_t(t) = \cos(wt)\cdot u(t) \cdot u(1 + \alpha - t)$ and $s_s(t) = \cos(wt)\cdot u(t - 1 - \alpha)$ but $s_s(t)$ does not have minimal bandwidth.
I would expect the above problem to have a unique solution. I imagine $s_s(t)$ have a Gaussian like attack. Any ideas?