Sequences of m-term deviations in Hilbert space

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 of x from 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 H 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.

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