  Dipole mixed pulsation modes of consecutive radial order have been detected
for thousands of low-mass red-giant stars with the NASA space telescope Kepler.
Such modes have the potential to reveal information on the physics of the deep
stellar interior. Different methods have been proposed to derive an observed
value for the gravity-mode period spacing, the most prominent one relying on a
relation derived from asymptotic pulsation theory applied to the gravity-mode
character of the mixed modes. Our aim is to compare results based on this
asymptotic relation with those derived from an empirical approach for three
pulsating red-giant stars. We developed a data-driven method to perform
frequency extraction and mode identification. Next, we used the identified
dipole mixed modes to determine the gravity-mode period spacing by means of an
empirical method and by means of the asymptotic relation. In our methodology,
we consider the phase offset, $\epsilon_{\mathrm{g}}$, of the asymptotic
relation as a free parameter. Using the frequencies of the identified dipole
mixed modes for each star in the sample, we derived a value for the
gravity-mode period spacing using the two different methods. These differ by
less than 5%. The average precision we achieved for the period spacing derived
from the asymptotic relation is better than 1%, while that of our data-driven
approach is 3%. Good agreement is found between values for the period spacing
derived from the asymptotic relation and from the empirical method.
  Full abstract in PDF file.
