  The transformation of canonical decay laws of moving unstable quantum systems
is studied by approximating, over intermediate times, the decay laws at rest
with superpositions of exponential modes via the Prony analysis. The survival
probability $\mathcal{P}_p(t)$, which is detected in the laboratory reference
frame where the unstable system moves with constant linear momentum $p$, is
represented by the transformed form $\mathcal{P}_0\left(\varphi_p(t)\right)$ of
the survival probability at rest $\mathcal{P}_0(t)$. The transformation of the
intermediate times, which is induced by the change of reference frame, is
obtained by evaluating the function $\varphi_p(t)$. Under determined
conditions, this function grows linearly and the survival probability
transforms, approximately, according to a scaling law over an estimated time
window. The relativistic dilation of times holds, approximately, over the time
window if the mass of resonance of the mass distribution density is considered
to be the effective mass at rest of the moving unstable quantum system.
