Sequences of [formula omitted]-term deviations in Hilbert space

Abstract

Let D be a dictionary in a Hilbert space H, that is, a set of unit elements whose linear combinations are dense in H. We consider the least m-term deviation σm(x) of an element x∈H: this is the distance from x to the set of all m-term linear combinations of elements of D. We prove a dichotomy result: for any dictionary D, either the sequence {σm(x)}m=0∞ decreases exponentially for every x∈H, or the rate of convergence σm(x)→0 can be arbitrarily slow. We seek universal dictionaries realizing all strictly decreasing null sequences as sequences of m-term deviations. All commonly used dictionaries turn out not to be universal. In particular, the least rational deviations in Hardy space H2 do not form certain strictly monotone null sequences. There are no universal dictionaries in finite dimensional Hilbert spaces. We construct a universal dictionary in every infinite dimensional Hilbert space.