Complete multipliers

Abstract:

We investigate whether the completeness of a complete orthonormal sequence for ${L^2}( - \pi ,\pi )$ is preserved if the sequence is perturbed by multiplying a portion of it by a fixed function. For the particular sequence $\{ {(2\pi )^{ - 1/2}}{e^{inx}}\}$ we show that given any $\psi \in {L^\infty }( - \pi ,\pi )$, except $\psi = 0$ a.e., there is a nontrivial portion of $\{ {(2\pi )^{ - 1/2}}{e^{inx}}\}$ which will maintain completeness under this perturbation.