On the Connection between Exponential Bases and Certain Related Sequences in L2(- Π,Π)

Abstract

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 ≤ Σn| ||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. © 1995 Academic Press, Inc.