On the Connection between Exponential Bases and Certain Related Sequences in L2(− π,π)

A sequence of vectors {fn} in a separable Hilbert space H is a frame if there are positive constants A, B such that A ||f||2 ≤ ∑n| 〈f, fn〉||2 ≤ B||f||2 for all f ∈ H, and {fn} is a Riesz sequence if it is a Riesz basis in the closure of the space spanned by the vectors fn. The latter of the following two questions has been raised by Khrushchev, Nikolskii, and Pavlov: Can every frame of complex exponentials {eiλnx} in L2 (− π, π) be made into a Riesz basis by removing from {eiλnx} a suitable collection of the functions eiλnx.; can every Riesz sequence {eiλnx} in L2 (−π, π) be made into a Riesz basis by adjoining to {eiλnx} a suitable collection of exponentials eiλnx ∉ {eiλnx}? We show that this is indeed so for all frames and Riesz sequences {eiλnx} studied so far, and we prove that we can always solve both problems in a certain "weak sense." However, our main conclusion is that the answer to both questions is no.