The standard answers

A straightforward solution to (1) is trace interpolation. But because of the very simple form of stretch, there are elegant alternatives to explicit trace interpolation.

After logarithmic and Fourier transformation of the time function p(t) the original stretch can be performed by a simple multiplication:  

$${P_{1}(\sigma_{1})} = {P(\sigma_{1}) e^{i \sigma_{1} \log(\alpha)}}$$ (2)

For brevity, we will refer to this ${\bf F}$ourier transformed ${\bf L}$ogarithmic domain as ${\bf FL}$ domain. $\sigma_{1}$ denotes the variable in the ${\bf FL}$ domain after stretching.

To prevent the time trace from being undersampled, the trace after logarithmic transformation will have to have n_s samples, where  

$${n_{s}} = {2 t_{\max} f_{\max} \log({t_{\max}\over{t_{\min}}})}.$$ (3)

$f_{\max}$ is the maximum frequency of p(t), $t_{\min}$, $t_{\max}$ are the upper and lower limits of the section of p(t), that the logarithmic trace of n_s points resamples correctly. In practice the logarithmic trace is often found to be extremely long.