Asteroseismology of DAV White Dwarfs with Time-Frequency Analysis and Band-Limited Interpolation

The study of pulsating white dwarfs proves to be an important source of fundamental astrophysical parameters, and may lead to constraints on the theory of stellar evolution. Among those stars, the ZZ Ceti, cool white dwarfs with pure Hydrogen atmosphere, are a very interesting group: their atmosphere is strongly stratified in pure Hydrogen and pure Helium layers; this phenomena induces trapping of the oscillations, and thus influences their frequency and amplitude. Asterosismology allows us to probe the content in Hydrogen and Helium, which have strong implications on the theory of stellar evolution and planetary nebulae. Long runs and multisite observations can yield also information on the rotation rate of the white dwarfs (by resolving the splitting of the non-radial modes). The coolest ZZ Ceti stars, as those we study in this paper, show very large oscillation amplitudes, and non stationary behaviour: as the amplitude of the modes can be changing in time, even in a single run of observation, classical spectral analysis is insufficient to disclose the rich behaviour of those objects. We introduce here time-frequency methods in order to improve the analysis of the observations.We propose to reconstruct temporal spectra obtained from incomplete light curves records of white dwarfs. The regularization of the underlying inverse problem makes reference to band-limited interpolation and matching pursuit. As the data in Fourier space are the result of a convolution operation, we try to reconstruct the spectrum that would have been obtained with a non-discontinuous or wider observing window function. Moreover, we search oscillations arising almost everywhere in the signal, which are characteristic of the structural properties of the star. So, it is necessary to get information about the time life corresponding to a given peak in the Fourier spectrum. The particular nature of this problem led us to time-frequency analysis, used as a complement to Fourier analysis, as a way of choosing these supports, and the matching pursuit algorithm plays a decisive role. This algorithm allows us to choose, in a given redundant finite dictionary of time- frequency waveforms, a set of atoms that match the signal as well as possible. A frequency support can therefore be defined for each coherent structure extracted presenting sufficiently long time life. The localization of each peak in the Fourier transform can be precisely read on the matching pursuit diagram. This allows us to detect coherent structures lost in the noise and then to choose a support over which a deconvolution procedure can be set in motion. This deconvolution procedure allows us, at the cost of a less full representati on of the spectrum, to formulate the deconvolution problem in terms of weighted temporal interpolation with partial extrapolation (band-limited interpolation). The corresponding regularization principle proves to be intimately related to the notion of resolution limit of the reconstruction process.Once the deconvolution is performed over one support detected with the Matching Pursuit algorithm, we can set in motion the deconvolution over the following one, and so on. And under the condition that the spectrum of the star can be broken up, we reconstruct it step by step, whereas a global reconstruction would be unstable at the same resolution. We illustrate this study with several application examples and we discuss its degree of confidence by comparing it with WET (Whole Earth Telescope) observations.