In the past 20 years, frames have become significant tools in a vast number of areas, both in research and in applications. However, sometimes it is not possible to find frames satisfying particular structural requirements.

In this talk a relaxed version of frames, which we call (QF)-systems, will be introduced. Roughly speaking, these are complete systems with an additional property - the coefficients of the approximating linear combinations are, in some sense, controlled. In different settings this concept enables us to get positive results which are known to be impossible for frames. More precisely, I will discuss the construction of sparse exponential systems {e^iλx} which are (QF)-systems in L²(S) for "generic" sets S of large measure. The Landau density theorem implies that it is impossible to construct such frames. In another setting, it is known that a system of translates {g(t-λ)}_{λ ∈ Λ} cannot be a frame in L²(S). On the other hand, it turns out that it is possible to construct a system of translates, with a sparse set of translations Λ, which is a (QF)-system in L²(R). Such a construction will be discussed as well.

This is a joint work with A.M.Olevskii.