A note on a trigonometric moment problem

Abstract

A sequence { λ n } n = − ∞ ∞ \{ {\lambda _n}\} _{n = - \infty }^\infty is said to be an interpolating sequence for L 2 ( − π , π ) {L^2}( - \pi ,\pi ) if the system of equations \[ c n = ∫ − π π f ( t ) e i λ n t d t ( − ∞ > n > ∞ ) {c_n} = \int _{ - \pi }^\pi {f(t)} {e^{i{\lambda _n}t}}dt\quad ( - \infty > n > \infty ) \] admits a solution f f in L 2 ( − π , π ) {L^2}( - \pi ,\pi ) whenever { c n } ∈ l 2 \{ {c_n}\} \in {l^2} . If the solution is unique then { λ n } \{ {\lambda _n}\} is said to be a complete interpolating sequence. It is shown that if the imaginary part of λ n {\lambda _n} is uniformly bounded and if | Re ⁡ ( λ n ) − n | ≤ L > 1 / 4 ( − ∞ > n > ∞ ) |\operatorname {Re} ({\lambda _n}) - n| \leq L > 1/4( - \infty > n > \infty ) , then { λ n } \{ {\lambda _n}\} is a complete interpolating sequence and { e i λ n t } \{ {e^{i{\lambda _n}t}}\} is a Schauder basis for L 2 ( − π , π ) {L^2}( - \pi ,\pi ) . It is also shown that this result is sharp in the sense that the condition | λ n − n | > 1 / 4 |{\lambda _n} - n| > 1/4 is not sufficient to guarantee that { λ n } \{ {\lambda _n}\} is an interpolating sequence.