Democratic systems of translates

Democratic systems arise in the context of greedy approximations in Banach spaces. The greedy algorithm is an important method in numerical mathematics; for more details see the early work of V. N. Temlyakov [16], [17]. At the same time it also points to some interesting issues in functional analysis. In particular, we would like to emphasize the basic theorem of S. V. Konyagin and V. N. Temlyakov (see [8]) which states that a basis in a Banach space is greedy if and only if it is unconditional and democratic. Hence, one could expect (and this is indeed so) that there are democratic bases which are not unconditional. It is important to mention that it is often difficult to produce valuable examples of conditional bases (it is good to consult books by I. Singer [15], J. Lindenstrauss and L. Tzafriri [9], R. Young [18]; numerous notions and fundamental results that we use here one can find there). We find an interesting direction in an article by P. Wojtaszczyk [19] where even more general systems than bases are studied in a similar context. The author studies quasi-greedy biorthogonal systems and develops a method to construct a wide range of quasi-greedy conditional bases. This partially motivates our work, since we shall also study that democratic property for general systems and we shall also try to offer deeper understanding of a particular, but fairly rich class of conditional democratic bases.