Sampling sets and sufficient sets for A−∞ ✩

We give new characterizations of the subsets S of the unit disc D of the complex plane such that the topology of the space A−∞ of holomorphic functions of polynomial growth on D coincides with the topology of space of the restrictions of the functions to the set S. These sets are called weakly sufficient sets for A−∞. Our approach is based on a study of the so-called (p, q)-sampling sets which generalize the A−p-sampling sets of Seip. A characterization of (p, q)-sampling and weakly sufficient rotation invariant sets is included. It permits us to obtain new examples and to solve an open question of Khôi and Thomas.  2003 Elsevier Science (USA). All rights reserved