Abstract:

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 *L*^{2}(*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 *L*^{2}(*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 *L*^{2}(**R**). Such a construction will be
discussed as well.

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