  A necessary and sufficient condition on the precompactness of extremal
sequences for one dimensional $\alpha$-Strichartz inequalities, equivalently
$\alpha$-Fourier extension estimates, is established based on the profile
decomposition arguments. One of our main tools is an operator-convergence
dislocation property consequence which comes from the van der Corput Lemma. Our
result is valid in asymmetric cases as well. In addition, we obtain the
existence of extremals for non-endpoint $\alpha$-Strichartz inequalities.
