  We consider the estimation of a common period for a set of functions sampled
at irregular intervals. The problem arises in astronomy, where the functions
represent a star's brightness observed over time through different photometric
filters. While current methods can estimate periods accurately provided that
the brightness is well--sampled in at least one filter, there are no existing
methods that can provide accurate estimates when no brightness function is
well--sampled. In this paper we introduce two new methods for period estimation
when brightnesses are poorly--sampled in all filters. The first, multiband
generalized Lomb-Scargle (MGLS), extends the frequently used Lomb-Scargle
method in a way that na\"{i}vely combines information across filters. The
second, penalized generalized Lomb-Scargle (PGLS), builds on the first by more
intelligently borrowing strength across filters. Specifically, we incorporate
constraints on the phases and amplitudes across the different functions using a
non--convex penalized likelihood function. We develop a fast algorithm to
optimize the penalized likelihood by combining block coordinate descent with
the majorization-minimization (MM) principle. We illustrate our methods on
synthetic and real astronomy data. Both advance the state-of-the-art in period
estimation; however, PGLS significantly outperforms MGLS when all functions are
extremely poorly--sampled.
