Some stability theorems for nonharmonic Fourier series

Abstract:

The theory of nonharmonic Fourier series in ${L^2}( - \pi ,\pi )$ is concerned with the completeness and expansion properties of sets of complex exponentials $\{ {e^{i{\lambda _n}t}}\}$. It is well known, for example, that the completeness of the set $\{ {e^{i{\lambda _n}t}}\}$ ensures that of $\{ {e^{i{\mu _n}t}}\}$ whenever $\sum {|{\lambda _n} - {\mu _n}| < \infty }$. In this note we establish two results which guarantees that if $\{ {e^{i{\lambda _n}t}}\}$ is a Schauder basis for ${L^2}( - \pi ,\pi )$, then $\{ {e^{i{\mu _n}t}}\}$ is also a Schauder basis whenever $\{ {\mu _n}\}$ is “sufficiently close” to $\{ {\lambda _n}\}$.